10: Positive and Negative Messages
Chapter Objectives
The purpose of this chapter is to:
- Present different types of routine messages
- Explain the difference between complaints and claims
- Identify parts of an adjustment message
- Explain the significance of apology
- Present different types of positive (goodwill) messages
- Discuss organization and impact of negative messages
Positive and Routine Messages
The vast majority of business messages sent every day are short messages of a routine nature such as asking for and sharing information, requesting action, or thanking someone for something given. Most of the time these are positive or neutral messages even when they involve small complaints or claims where you request that an error be corrected. These are all direct approach messages , in which the main idea comes right upfront and details follow. Occasionally, you must communicate bad news in writing, which requires a more careful, indirect approach. All of these situations involve conventions that business professionals follow to minimize miscommunication and its fallout, and thus keep their operation running smoothly.
Positive (Goodwill) Messages
Goodwill messages are as essential to healthy professional relationships as they are to personal ones. Thank-you, congratulatory, and sympathy notes add an important, feel-good human touch in a world that continues to embrace technology that isolates people while being marketed as a means of connecting them. The goodwill that such messages promote makes both the sender and receiver feel better about each other and themselves. In putting smiles on faces, such notes are effective especially because many people don't send them--either because they feel that they are too difficult to write or because it does not even occur to them to do so. Since praise for someone can be harder to think of and write than criticism, a brief guide on how to do it right may be of help .
| Type of Message | Characteristics |
| Thank-You Notes | Such messages should be short and simple as well as quick and easy to write. For that reason, not sending them when someone does something nice for you appears ungrateful, rude, and inconsiderate. In fact, sending a thank-you note within 24 hours of interviewing for a job, for example, is not just extra thoughtful but close to being an expected formality. In most situations, email or text is an appropriate medium for sending thank-you messages. |
| Congratulatory Messages | Celebrating the successes of your professional peers shows class and tact. Such messages create positive energy in the workplace and should be all about he person you are congratulating. |
| Expressions of Sympathy |
Few situations require such sincerity and care with words as expressions of sympathy. Misfortune comes upon us all, and tough times are just a little more tolerable with the support of our friends, family, and community--including those we work with. When the loved-one of a close associate dies, for instance, expressing sympathy for their loss is customary, often with a card, rather than an email or text, signed by everyone in the workplace who knows the bereaved. You can't appreciate email messages the same way you can a collection of cards from people showing they care. Remember that these messages should be selfless. Offering your condolences in the most respectful, sensitive manner possible is the right thing to do. |
Information Shares
Perhaps the simplest and most common routine message type is where the sender offers up information that helps the receiver .
Information Share Email
To<EMAIL_ADDRESS>Subject: CFP info you might find interesting
Hi Karin,
I just saw a CFP for a new funding opportunity you can apply for via the Department of Agriculture.
Find it on the Greenbelt Fund's Local Food Literacy Grant Stream page. If you haven't already been doing this, you should also check out the Department of Agriculture's general page on Funding Programs and Support to connect with any other grants etc. relevant to the good work you do.
It looks like the deadline for proposals is at the end of the week, though, so you might want to get on it right away.
Good luck!
Rima
Replies to such information shares involve either a quick and concise thank-you message or carry the conversation on if it's part of an ongoing project, initiative, or conversation. If you are using email, remember to change the subject line as the topic evolves. Information shares to a larger group, such as a departmental memo to 60 employees, don't usually require acknowledgment and would be slightly more formal in tone. If everyone wrote the sender just to say thanks, the barrage of reply notifications would frustrate them as they try to carry on their work while sorting out replies with valuable information from mere acknowledgments. Only respond if you have valuable information to share with all the recipients or just the sender. Pay attention whether you need to Reply All or whether it is sufficient to reply just to the sender. Do not burden others with messages not directly relevant to them.
Information or Action Requests
Managers, clients, and coworkers alike send and receive requests for information and action all day. Because these provide the recipient with direction on what to do, the information that comes back or action that results from such request can only be as good as the instructions given. Such messages must be well organized and clear about expectations, opening directly with a clearly stated general request (unless you anticipate resistance to the request) and proceeding with background and more detailed instructions if necessary. Figure 1 below provides an example of a request email.
Figure 1: Request Email, Communication @ Work
Note that, because we expect action to come of the request rather than a yes or no answer, the opening question does not require a question mark. Never forget, however, the importance of saying "please" when asking someone to do something. Notice also that the list in the message body helps break up dense detail, so the request message is more reader friendly. All of the effort that the writer of the above message made to deliver a reader-centered message will pay off when the recipient performs the requested procedure exactly according to these clearly worded expectations.
If you expect resistance to your request, an indirect approach is more effective. Ideally, you'll make such persuasive pitches in person or on the phone so that you can use a full range of verbal and non-verbal cues. When it's important to present your argument in writing, however, such requests should be clear and easy to spot but buffered by goodwill statements and reasonable justifications.
When responding to information or action requests, simply deliver the needed information or confirm that the action has been or will be completed unless you have good reasons for refusing. Ensure the accuracy of all details and utilize the "you-attitude." Note that sometimes manager approval may be necessary before providing the information. Organizationally, a positive response to an information request delivers the main answer in the opening, proceeds to give more detail in the body if necessary, and ends politely with appreciation and goodwill statements.
Complaints and Claims
Business doesn't always go smoothly and bad customer experience can have long-lasting consequences. Customers can be disappointed with a faulty product or poor service; shipments might get damaged, lost, or arrive late; or one business might infringe on the rights and freedoms of another. In all such cases, customers or clients are likely to make your company aware of what went wrong and what they want o be done about it. Indeed, it's their consumer right to do so and the business or organization receiving such a message should take it as valuable intelligence on customer expectations that must be met for the business to be viable.
A claim explains what went wrong and demands compensation from the offending party while a complaint explains what went wrong and merely demands correction or apology. Minor complaints are best communicated in person, on the phone, or by email (if it's important to have them in writing) so they can be dealt with quickly. More serious complaints or claims are delivered as formal letters to lay down a paper trail in case they need to be used as evidence in a lawsuit.
When customers are reasonable about communicating a problem with a situation or business transaction, the customer service representative or manager dealing with the matter is more likely to respond positively and meet the needs of the client. However, ineffective complaints or claims often merely vent frustrations, issue threats, don'ts say what they want or only vaguely imply it, or demand completely unreasonable compensation. Assume that a business will take your complaint or claim seriously if it's done right because, no matter the industry, companies are rightfully afraid of losing business to negative online reviews. In this day of social media, good customer service is crucial to business survivability. A complaint provides a business with both valuable information about customer expectations and an opportunity to win back a customer--as well as their social network if a good endorsement comes of it from the now-satisfied customer--or else risk losing much more than just the one customer.
Effective complaints or claims are politely worded and motivated by a desire to right wrongs and save the business relationship. They are best if they remind the business that you have been a loyal customer (if that's true) and really want to keep coming back, but you need them to prove that they value your business after whatever setback prompted the complaint. If the writer of such message strikes the right tone, they can end up getting more than they originally bargained for.
Complaints and claims take the direct approach of message organization even though they arise from dissatisfaction. They follow the usual three-part message organization .
Complaint/Claim Organization
- Opening - To be effective at writing a complaint or claim, be clear, precise, and polite about what you want in the opening. If you want financial compensation or a replacement product in the case of a claim, be clear about the amount or model. You could also suggest equivalent or alternative compensation if you stand a poor chance of getting exactly what you want. If you want an error corrected or an apology in response to your complaint, be upfront about it.
- Body - The message body justifies the request with a narrative account of what should have happened versus what actually happened instead. Be objective in writing the account because an angry tone coming through in negative words, accusations, and exaggerations will only undermine the validity of your complaint or claim. Be precise in such details as names, dates and times, locations (addresses), and product names and numbers. Whenever possible, provide and refer to evidence. For instance, you may include copies (definitely not originals) of documentation such as receipts, invoices, work orders, bills of lading, emails (printed), phone records, photographic evidence, and even video (e.g., of a malfunctioning product).
- Closing - No matter what prompted the complaint or claim, the closing must be politely worded with action requests (e.g., a deadline) and goodwill statements. Nasty parting shots, even if merely passive-aggressive, may lower your chances of getting what you are asking for. By complimenting the recipient's company, however, you increase your chances of getting not only what you wanted, but perhaps a little extra. In damage-control mode, the business wants you to feel compelled to tell your friends that the company really turned it around.
Claim Email
To<EMAIL_ADDRESS>Subject: Refund for unwanted warranty purchase
Hello,
please refund me for the $89.99 extended warranty that was charged to my Visa despite being declined at the point of sale.
This past Tuesday (June 15, 2021), I purchased a Dell laptop at the Belleville location of Future Shock Computers and was asked by the sales rep if I would like to add a 3-year extended warranty to the purchase. I declined and we proceeded with the sale, which included some other accessories. When I got home and reviewed the receipt (the PDF scan of the receipt is attached), I noticed the warranty that I had declined was added to the bill after all.
Please apply the cost of the warranty to the Visa account associated with the purchase by the end of the week and let me know when you've done so. I have enjoyed shopping at Future Shock for the great prices and friendly customer service. I would sincerely like to return to purchase a printer soon.
I appreciate your assistance in this matter,
Samantha Nguyen
Adjustment Letters
If a company grants what the customer has asked for, communicating this is called an adjustment message. An adjustment letter or email is heavy on courtesy in letting the disappointed customer know that they are valued and will be (or have already been) awarded what they were asking for, and possibly even a little extra. In the case of coupons for discounts on future purchases, the little extras help smooth things over and win back the customer's confidence, hopefully so they will tell their friends that the store or company is worthy of their business after all.
An adjustment message takes the direct approach by immediately delivering the good news about granting the customer's request. You may start with an apology, but starting on a purely positive note is more effective, especially in a written message. Tone is also important here; resist the urge to shame the customer--even if they are partly to blame or if part of you still suspects that the claim is fraudulent. If you are going to grant the claim, write it whole-heartedly as if others will be able to see it and judge whether your company has good customer service or if you are going to be difficult about it.
Though a routine adjustment letter might skip a message body, a more serious one may need to go into more detail about how you are complying with the request or take the time to explain what your company is doing to prevent the error again. Doing this makes the reader feel as though making the effort to write will have made a positive impact in the world, however small, because it will benefit not only them but also everyone else who will not have to go through what they did. Do not make empty promises, though.
An apology might also be appropriate in the body of the message, but be aware that apologizing is tricky. It is essential to winning back customer confidence in some situations, but it also leaves you or your company open to legal action in others. For minor matters, admitting fault with an apology usually helps vindicate or validate the customer. In more serious matters, especially involving injury or damage to property or even someone's reputation (and thus their earning potential), a written apology might be read as admitting fault and be used as evidence in court. For this reason, it is best to ask a manager or legal department for guidance on apologizing to a customer or other stakeholder in writing.
If apologizing is appropriate because you genuinely erred, no legal repercussions are expected, and it is the right thing to do when trying to calm an angry response and mend a damaged business relationship, ensure that the apology has the following characteristics:
- Sincere - Saying "We are genuinely sorry that you were disappointed with the customer service experience" is a good first step, but it requires some additional assurances to prove it.
- Responsible - Own the error by admitting fault (again, only if you are at fault and it does not open you to litigation). Say what should have happened versus what actually happened and acknowledge that you were wrong.
- Specific - To be sincere, an apology must refer to a specific error by briefly describing it, possibly including dates, locations, and the names of product or service. The worst apologies are blanket, generic statements such as "We are sorry if anyone was offended by our actions." On the other hand, it is unprofessional to throw your employees under the bus or to vent about the situation in your company that might have contributed to the problem. These issues should be dealt with within the business.
- Improvement-focused - An apology is useless unless it includes some assurances that you will make every effort for the error to not happen again. Simply saying it won't happen again is not convincing and might sound as an empty promise. Instead, describe what will be done to ensure improvement. When Starbucks apologized for a racist incident at one of its locations in 2018, for instance, it followed through on a plan for improving customer service by shutting down all of its stores for a half-day racial sensitivity training for employees .
Adjustment Email
Subject: Refund for unwanted warranty purchase
Hello Samantha,
absolutely, we would be happy to refund you for the $89.99 warranty mistakenly charged along with your purchase of the Dell laptop. For your inconvenience, we will also offeer you a $20 gift card for future purchases at our store.
To receive your refund and gift card, please return to our Belleville location with your receipt and the credit card that was charged so that we can credit the same card (For consumer protection reasons, we are unable to complete any transactions without the card.)
We are sorry for inconveniencing you and will speak with all sales staff about the importance of carefully checking the accuracy of any bill of sale before sending the order for payment.
We appreciate your choosing Future Shock for your personal electronics and look forward to seeing you soon to credit your Visa card and provide you with the best deal in town on the printer you were looking to purchase.
Have a great day,
Melissa
Negative Messages
Just as in life, the workplace is not always sunny. Sometimes things do not go according to plan, and it is your job to communicate about them in a way that does not ruin your relationships with customers, coworkers, managers, the public, and other stakeholders. When doing damage control, bad-news messages required care, diplomacy, and skillful language because your main point will meet resistance.  Though some people prefer that the messenger be blunt about it, in most cases you can assume that the receiver will appreciate or even benefit from a more tactful, indirect approach. The list below provides several tips for delivering bad news in person or in writing:
- Be clear and concise to avoid being asked for additional clarification.
- Help the receiver understand and accept the news.
- Reduce the anxiety associated with the bad news as much as possible by expressing sympathy or empathy.
- Maintain trust and respect between you and your audience to ensure the possibility of good future relations.
- Deliver the bad news in a timely fashion through the appropriate medium.
- Avoid the legal liability that comes with admitting negligence or guilt.
- Achieve the designated business outcome.
For more information on delivering unwelcome news effectively, watch Don R. Crawley, IT Customer Service Speaker, in this video: How to Deliver Bad News: Customer Service Training 101 , which provides five strategies for delivering bad news. Your ability to manage, clarify, and guide understanding is crucial to addressing challenging situations while maintaining trust and integrity.
Key to helping the receiver understand and accept information they don't want to hear or read is organizing the message using the indirect approach . For example, this is effective when you need to refuse a claim because the customer's complaint is not justified. If you tactlessly provide your audience with really bad news, you run the risk of them rejecting or misunderstanding it because they may be reeling from the blow and be too distracted with anger or sadness to rationally process the explanation or instructions for what to do about the bad news. To avoid misunderstandings when delivering bad news, follow this organization .
Bad News Message Organization
- Buffer (Acknowledgment of the Claim) - Begin with a neutral or positive statements that set a goodwill tone and serve as a buffer for the information to come. A buffer softens the blow of bad news. The idea here is not to fool the audience into thinking that only good news is coming but to put them in a receptive frame of mind for understanding the explanation that follows. Note that if you raise the expectation that they are going to hear the good news that they are getting what they want, only to let them down near the end, they are going to be even more disappointed for being led on. For that reason, be careful not to create false hope.
- Justification (Explanation) - The justification explains the background or context for the bad news before delivering the bad news itself. Let's say that you must reject an application, claim for a refund, or request for information. In such cases, the explanation could describe the strict acceptance criteria and high quality of applications received in the competition, the company policy on refunds, or its policy on allowable disclosures and the legalities of contractually obligated confidentiality. Your goal with the explanation is to be convincing so that the reader thinks, "That sounds reasonable" or "I understand their point" and accepts the bad news as inevitable given the situation you describe. If you make the bad news seem like mysterious or arbitrary decision making, your audience will most likely feel like they've been treated unfairly. Such situations usually escalate further, which we always try to avoid. While an explanation is ethically necessary, never admit or imply responsibility without written authorization from your company cleared by legal counsel if there is any indication that the justification might be seen as actionable.
- Bad News - We usually use the indirect approach to delay the bad-news message so that it can be properly understood and its negative impact minimized. Be clear while also being sensitive to your reader's feelings. Consider using passive voice when delivering this news.
- Redirection (Alternative) - This is keBay to bad news' effectiveness because it quickly shifts the reader's attention to an alternative to what they are seeking. Some kind of consolation prize (e.g., a coupon or store credit) helps soothe the pain and will be appreciated as being better than nothing, at least. Even if you are not able to offer the reading anything of value, you could at least say something nice or provide some useful information. If your redirection involves some alternative, remember to also offer some follow-up details. For example, if you are offering coupon or store credit, explain how the customer should redeem this offer.
- Positive Closing - The goal of the closing is to leave on a positive note. You need to ensure that the reader understands the bad news without rehashing it. Remain courteous, positive, and forward looking. End the conversation in such a way that you do not invite further communication unless it is necessary with respect to the alternative that you have offered .
Bad News Message Outline and Example
| Part | Example Message |
|---|---|
| 1. Buffer | Thank you for your order. We appreciate your interest in our product and are confident you will love it. |
| 2. Explanation | We are writing to let you know that this product has been unexpectedly popular with over 10,000 orders submitted on the day you placed yours. |
| 3. Bad news + redirect | This unexpected increase in demand has resulted in a temporary out-of-stock/backorder situation. Despite a delay of 2-3 weeks, we will definitely fulfill your order as it was received at 11:57 p.m. on October 9, 2018, as well as gift you a $5 coupon towards your next purchase. |
| 4. Positive action closing | While you wait for your product to ship, we encourage you to use the enclosed $5 coupon toward the purchase of any product in our online catalog. We appreciate your continued business and want you to know that our highest priority is your satisfaction. |
Although it is usually preferable to deliver negative messages using the indirect approach, there are several occasions when you can or should be upfront about the bad news:
- Wen the bad news does not have a high emotional impact
- When you know that the recipient prefers or requires the direct approach
- When the reader may miss the bad news (e.g., if your reader doesn't have a strong command of English, they may not pick up on the buried bad news past the midpoint of a challenging message)
- When you are short on time or space
- When the indirect approach has not worked.
Delivering bad news can be dangerous if it angers the reader so much that they are motivated to fight back. If you are not careful with what you say and how you say it, that message can be used as evidence in a court case that, when read by a judge or jury, could compromise your position. You can lower the risk of being litigated by following general principles given below when delivering bad news.
- Avoid negative or abusive language - sarcasm, profanity, harsh accusations, and abusive or insulting language may feel good to write in a fit of anger but, in the end, make everyone's lives mroe difficult. When someone sends an inflammatory message and it's interpreted by the reader as harmful to their reputation, it could legally qualify as libel that is legitimately actionable. Even if you write critically about a rival company's product or service by stating (as if factually) that it's dangerous, whereas your version of the product or service is safer and better, this can be considered defamation or libel. If said aloud and recorded, perhaps on a smart phone's voice recorder, ti is slander and can likewise be litigated. It is much better to always write courteously and maturely, even under difficlut circumstances, to avoid fallout that involves expensive court proceedings.
- Avoid oversharing but tell the truth - When your job is provide a convincing rationale that might make the recipient of bad news accept it as reasonable, be careful with what details you disclose. When rejecting a job applicant, for instance, you must be especially careful not to share the scoring sheets of the winning and rejected candidates, nor even summarize them. Though that would give them full picture, it would open you up to a flood of complaints and legal challenges. Instead, you should simply wish the rejected candidate luck in their ongoing job search. When you must provide detail, avoid saying anything bad about anyone so that you can't be accused of libel and taken to court for it. Provide only as much information as necessary to provide a convincing rationale. At the same time, it is important to tell the truth. If you are inconsistent or contradictory in your explanation, it may invite scruity and accusations of lying. Even making false claims by exaggerating my give the reader the wrong impression, which can lead to serious consequences. Though some might say that omitting the truth is a form of lying, telling the truth selectively is the necessary compromise of a professional constrained by competing obligations to both the organization they represent and the reader whom they don't want to anger or severely disappoint.
- Respect the recipient's privacy - criticizing an employee in a group email or memo--even if the criticism is fair--is extremely unprofessional. People who call out others in front of a group create a chilly climate in the workplace, one that leads to fear, loathing, and a loss of productivity among employees. Always maintain respect and privacy when communicating bad news as a matter of workplace professionalism.
General Positive and Negative Messages Tips
- Follow best practices when sharing information, requesting information or action, and replying to such messages.
- Be courteous both when complaining and responding to complaints and claims.
- Write carefully when addressing negative situations, such as delivering bad news, usually by using the indirect approach.
All links live as of June 2021.
This work "Positive and Negative Messages" is a derivative of “ Communication @ Work ” by Tom Bartsiokas, Robin Potter, and Tricia Hylton under a CC BY license. "Postive and Negative Messages" is licensed under CC BY 4.0 by Iva Balic.


14.4: Important Examples
We can use a graph to visualize the power set of a finite set \(A\text{:}\) let \((\mathscr{P}(A),E)\) be the directed graph where, for vertices \(B,C \in \mathscr{P}(A)\) (that is, subsets \(B,C \subseteq A\)), the ordered pair \((B,C)\) is an edge in \(E\) if the following two conditions are satisfied:
\(B \subsetneqq C\text{;}\) and
there does not exist a subset \(D \subseteq A\) such that \(B \subsetneqq D \subsetneqq C\text{.}\)
Note: The second condition is to avoid cluttering the graph with extra edges that do not add any extra information.
For example, consider \(A = \{a,b,c\}\) and
\begin{equation*} \mathscr{P}(A) = \left\{ \, \emptyset, \, \{a\}, \, \{b\}, \, \{c\},\, \{a,b\},\, \{a,c\},\, \{b,c\},\, \{a,b,c\} \, \right\}. \end{equation*}
So we can draw a graph to represent the subset relationships of \(\mathscr{P}(A)\text{,}\) with arrows to point from subset to superset.
It is somewhat natural to draw the graph with the largest subsets of \(A\) at the top (or at the bottom). If we decide that arrows will always point upwards, we can unclutter our graph by just drawing lines instead of arrows for edges.
We can use graphs to visualize other kinds of mathematical relationships.
For integers \(m,n\text{,}\) write \(m \mid n\) if \(m\) divides \(n\text{;}\) that is, if \(n/m\) is also an integer. In this case, we say that \(m\) divides \(n\text{.}\)
Set
\begin{equation*} V = \{2,3,4,5,6,7,8,9,10,11,12,13,14\} \text{.} \end{equation*}
Let \(G = (V,E)\) be the directed graph where, for integers \(m,n \in V\) with \(m \lt n\text{,}\) the ordered pair \((m,n)\) is an edge in \(E\) if the following two conditions are satisfied:
\(m \mid n\text{;}\)
there does not exist an integer \(\mathscr{P} \in V\) such that \(m \mid \mathscr{P}\) and \(\mathscr{P} \mid n\text{,}\) but \(\mathscr{P} \ne m,n\text{.}\)
We can use this directed graph to visualize the set \(V\) with respect to the divides relationship. Again, we agree that arrows will always point up, but do not actually draw them. Note that the prime numbers all appear at the bottom.
You get a summer job with the Prospective Student Office at the University of Alberta's Augustana Campus in Camrose, Alberta. In May and June, your job is to visit Alberta high schools and meet with students who are thinking of applying to Augustana. Below is a map of the cities and towns you must visit, given as a weighted graph with distances in kilometres as weights. To save on gas, you would like to visit each city and town exactly once, and do so while travelling the shortest distance possible.
This is a difficult problem, and gets more difficult as the number of cities and roads increases.


6.2: Enzyme Kinetics
Enzymes are catalysts, usually proteins, that help convert other molecules called substrates into products, but are themselves unchanged by the reaction. Each enzyme has high specificity for at least one reaction, and it can accelerate this reaction by millions of times.  Enzymes are so important to our lives that a single amino acid mutation in one enzyme out of the more than 2000 enzymes in our bodies can result in a severe or lethal genetic disease.
where \(C\) is a complex formed by the enzyme and the substrate. A cartoon of the Michaelis-Menten reaction with an enzyme catalyzing a reaction between two substrates is shown in Fig. 6.1. Commonly, substrate is continuously provided to the reaction and product is continuously removed. The removal of product has been modeled by neglecting the reverse reaction \(P+E \rightarrow C\) . A continuous provision of substrate allows us to assume that \(S\) is held at an approximately constant concentration.
The differential equations for \(C\) and \(P\) can be obtained from the law of mass action:
\[\begin{aligned} &d C / d t=k_{1} S E-\left(k_{-1}+k_{2}\right) C \\[4pt] &d P / d t=k_{2} C . \end{aligned} \nonumber \]
Biochemists usually want to determine the reaction velocity \(d P / d t\) in terms of the substrate concentration \(S\) and the total enzyme concentration \(E_{0}\) . We can eliminate \(E\) in favor of \(E_{0}\) from the conservation law that the enzyme, free and bound, is conserved; that is
\[\frac{d(E+C)}{d t}=0 \quad \Longrightarrow \quad E+C=E_{0} \quad \Longrightarrow \quad E=E_{0}-C \nonumber \]
and we can rewrite the equation for \(d C / d t\) eliminating \(E\) :
\[\begin{align} \nonumber \frac{d C}{d t} &=k_{1} S\left(E_{0}-C\right)-\left(k_{-1}+k_{2}\right) C \\[4pt] &=k_{1} E_{0} S-\left(k_{-1}+k_{2}+k_{1} S\right) C \end{align} \nonumber \]
Because \(S\) is assumed to be held constant, the complex \(C\) is expected to be in equilibrium, with the rate of formation equal to the rate of dissociation. With this so-called quasi-steady-state approximation, we may assume that \(C=0\) in \((6.3.4)\) , and we have
\[C=\frac{k_{1} E_{0} S}{k_{-1}+k_{2}+k_{1} S} \nonumber \]
The reaction velocity is then given by
\[\begin{align} \nonumber \frac{d P}{d t} &=k_{2} C \\[4pt] \nonumber &=\frac{k_{1} k_{2} E_{0} S}{k_{-1}+k_{2}+k_{1} S} \\[4pt] &=\frac{V_{m} S}{K_{m}+S} \end{align} \nonumber \]
where two fundamental constants are defined:
\[K_{m}=\left(k_{-1}+k_{2}\right) / k_{1}, \quad V_{m}=k_{2} E_{0} . \nonumber \]
The Michaelis-Menten constant or the Michaelis constant \(K_{m}\) has units of concentration, and the maximum reaction velocity \(V_{m}\) has units of concentration divided by time. The interpretation of these constants is obtained by considering the following limits:
\[\begin{array}{ll} \text { as } S \rightarrow \infty, & C \rightarrow E_{0} \text { and } d P / d t \rightarrow V_{m} \\[4pt] \text { if } S=K_{m}, & C=\frac{1}{2} E_{0} \text { and } d P / d t=\frac{1}{2} V_{m} . \end{array} \nonumber \]
Therefore, \(V_{m}\) is the limiting reaction velocity obtained by saturating the reaction with substrate so that every enzyme is bound; and \(K_{m}\) is the concentration of \(S\) at which only one-half of the enzymes are bound and the reaction proceeds at one-half maximum velocity.


1.1.3: Venn Diagrams
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To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the 18 th century. These illustrations now called Venn Diagrams .
Venn Diagram
A Venn diagram represents each set by a circle, usually drawn inside of a containing box representing the universal set. Overlapping areas indicate elements common to both sets.
Basic Venn diagrams can illustrate the interaction of two or three sets.
Example 9
Create Venn diagrams to illustrate \(A \cup B, A \cap B,\) and \(A^{c} \cap B\)
\(A \cup B\) contains all elements in either set.
\(A \cup B\) contains all elements in either set.
\(A \cap B\) contains only those elements in both sets - in the overlap of the circles.
Example 10
Use a Venn diagram to illustrate \((H \cap P)^{c} \cap W\)
We'll start by identifying everything in the set \(\mathrm{H} \cap P\)
Now, \((H \cap P)^{c} \cap W\) will contain everything not in the set identified above that is also in set \(W\)
Example 11
Create an expression to represent the outlined part of the Venn diagram shown.
The elements in the outlined set are in sets \(\mathrm{H}\) and \(F\), but are not in set \(W\). So we could represent this set as \(H \cap F \cap W\)
Try it Now 3
Create an expression to represent the outlined portion of the Venn diagram shown
- Answer
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\(A \cup B \cap C^{c}\)


27.5: Terpenoids
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- Steven Farmer, Dietmar Kennepohl, Tim Soderberg, & William Reusch
- University of Illinois Springfield
Objectives
After completing this section, you should be able to
- identify a terpene from a given list of organic structures.
- analyze the structure of a given terpene in terms of the isoprene rule.
- classify a given terpene structure according to the number of isoprene units present; that is, determine whether a given terpene is a monoterpene, sesquiterpene, diterpene, etc.
- identify and draw the structure of the precursors of isopentenyl diphosphate: ( R ) mevalonate or 1‑deoxy‑D‑xylulose 5‑phosphate.
Key Terms
Make certain that you can define, and use in context, the key terms below.
- isoprene rule
- terpenoid
- terpene
Study Notes
You are not expected to memorize all the details of the synthetic mechanisms for terpenoids. However, you should know the overall general synthetic pathway illustrated under “Terpenoid Biosynthesis.” As you read through the details of these mechanisms, realize that they may be complex, but they are based on experimental evidence. You should also note the important role of enzymes in many natural systems transformations. Finally, you will recognize that essentially, individual steps are often reactions you have already encountered in previous sections.
The terpenoids (aka isoprenoids) are a large (estimated 60% of known natural products ) and diverse group of lipids derived from five-carbon isoprene units assembled in thousands of combinations. Technically a terpenoid contains oxygen, while a terpene is a hydrocarbon. Often the two terms are used to refer collectively to both groups.
Isoprene Rule
Compounds classified as terpenes constitute what is arguably the largest and most diverse class of natural products. A majority of these compounds are found only in plants, but some of the larger and more complex terpenes (e.g. squalene & lanosterol) occur in animals. Terpenes incorporating most of the common functional groups are known, so this does not provide a useful means of classification. Instead, the number and structural organization of carbons is a definitive characteristic. Terpenes may be considered to be made up of isoprene (more accurately isopentane) units, an empirical feature known as the isoprene rule . Because of this, terpenes usually have \(5n\) carbon atoms (\(n\) is an integer), and are subdivided as follows:
| Classification | Isoprene Units | Carbon Atoms |
|---|---|---|
| monoterpenes | 2 | C 10 |
| sesquiterpenes | 3 | C 15 |
| diterpenes | 4 | C 20 |
| sesterterpenes | 5 | C 25 |
| triterpenes | 6 | C 30 |
Isoprene itself, a C 5 H 8 gaseous hydrocarbon, is emitted by the leaves of various plants as a natural byproduct of plant metabolism. Next to methane it is the most common volatile organic compound found in the atmosphere. Examples of C10 and higher terpenes, representing the four most common classes are shown in the following diagrams. Most terpenes may be structurally dissected into isopentane segments. How this is done can be seen in the diagram directly below.
The isopentane units in most of these terpenes are easy to discern, and are defined by the shaded areas. In the case of the monoterpene camphor, the units overlap to such a degree it is easier to distinguish them by coloring the carbon chains. This is also done for alpha-pinene. In the case of the triterpene lanosterol we see an interesting deviation from the isoprene rule. This thirty carbon compound is clearly a terpene, and four of the six isopentane units can be identified. However, the ten carbons in center of the molecule cannot be dissected in this manner. Evidence exists that the two methyl groups circled in magenta and light blue have moved from their original isoprenoid locations (marked by small circles of the same color) to their present location. This rearrangement is described in the biosynthesis section. Similar alkyl group rearrangements account for other terpenes that do not strictly follow the isoprene rule.
Polymeric isoprenoid hydrocarbons have also been identified. Rubber is undoubtedly the best known and most widely used compound of this kind. It occurs as a colloidal suspension called latex in a number of plants, ranging from the dandelion to the rubber tree ( Hevea brasiliensis ). Rubber is a polyene, and exhibits all the expected reactions of the C=C function. Bromine, hydrogen chloride and hydrogen all add with a stoichiometry of one molar equivalent per isoprene unit. Ozonolysis of rubber generates a mixture of levulinic acid ( \(CH_3COCH_2CH_2CO_2H\) ) and the corresponding aldehyde. Pyrolysis of rubber produces the diene isoprene along with other products.
The double bonds in rubber all have a Z-configuration, which causes this macromolecule to adopt a kinked or coiled conformation. This is reflected in the physical properties of rubber. Despite its high molecular weight (about one million), crude latex rubber is a soft, sticky, elastic substance. Chemical modification of this material is normal for commercial applications. Gutta-percha (structure above) is a naturally occurring E-isomer of rubber. Here the hydrocarbon chains adopt a uniform zig-zag or rod like conformation, which produces a more rigid and tough substance. Uses of gutta-percha include electrical insulation and the covering of golf balls.
Terpenoid Biosynthesis
While we can identify isoprene units within a terpenoid structure and use that in its classification, the building block for terpenoid synthesis in nature is isopentenyl diphosphate (formerly called isopentenyl pyrophosphate and abbreviated IPP). There are two major routes to the synthesis of IPP; namely (1) the mevalonate pathway and (2) the 1-deoxyxylulose pathway.
Mevalonate Pathway
Step 1 - Claisen Condensation
An early step in the biosynthesis of cholesterol and other ‘isoprenoid’ compounds is a Claisen condensation between two acetyl CoA molecules. An initial trans-thioesterase process transfers the acetyl group of the first acetyl CoA to an enzymatic cysteine (Reaction 1). In the Claisen condensation phase of the reaction, the alpha-carbon of a second acetyl CoA is deprotonated, forming an enolate (Reaction 2).
The enolate carbon attacks the electrophilic thioester carbon, forming a tetrahedral intermediate (Reaction 3) which quickly collapses to expel the cysteine thiol (Reaction 4) and produce acetoacetyl CoA.
Step 2 - Aldol Condensation
Acetyl CoA then reacts with the acetoacetyl CoA in an aldol-like addition. Subsequent hydrolysis produces (3 S )-3-hydroxy-3-methylglutaryl CoA (HMG-CoA).
Generating HMG-CoA
Step 3 - Reduction of the Thioester
The thioester is reduced first to an aldehyde, then to a primary alcohol by two equivalents of NADPH producing ( R )-mevalonate. The enzyme catalyzing this reaction is the target of the statin family of cholesterol-lowering drugs.
Generating ( R )-Mevalonate
Step 4 - Mevalonate Phosphorylation
Two phsophorylations by adenosine triphosphate (ATP) occur at the terminal hydroxyl/phosphorus group through nucleophilic substitution, followed by a third ATP phosphorylation of the tertiary hydroxyl group.
Step 5 - Decarboxylation
Finally isopentenyl diphosphate (IPP), the 'building block' for all isoprenoid compounds, is formed from a decarboxylation-elimination reaction.
Conversion of IPP to Terpenoids
The electrophilic double bond isomerization catalyzed by IPP isomerase is a highly reversible reaction, with an equilibrium IPP:DMAPP ratio of about 6:1. In the next step of isoprenoid biosynthesis, the two five-carbon isomers condense to form a 10-carbon isoprenoid product called geranyl diphosphate (GPP).
This is a nice example of an electrophilic addition/elimination mechanism:
The first step is ionization of the electrophile - in other words, the leaving group departs and a carbocation intermediate is formed. In this case, the pyrophosphate group on DMAPP is the leaving group, and the electrophilic species is the resulting allylic carbocation.
In the condensation (addition) step, the C 3 -C 4 double bond in IPP attacks the positively-charged C 1 of DMAPP, resulting in a new carbon-carbon bond and a second carbocation intermediate, this time at a tertiary carbon. In the elimination phase, proton abstraction leads to re-establishment of a double bond in the GPP product. Notice that the enzyme specifically takes the pro-R proton in this step.
To continue the chain elongation process, another IPP molecule can then condense, in a very similar reaction, with C 1 of geranyl diphosphate to form a 15-carbon product called farnesyl diphosphate (FPP).
How do we know that these are indeed S N 1-like mechanisms with carbocation intermediates, rather than concerted S N 2-like mechanisms? First of all, recall that the question of whether a substitution is dissociative (S N 1-like) or associative (S N 2-like) is not always clear-cut - it could be somewhere in between, like the protein prenyltransferase reaction . The protein prenyltransferase reaction and the isoprenoid chain elongation reactions are very similar: the electrophile is the same, but in the former the nucleophile is a thiolate, while in the latter the nucleophile is a pi bond.
This difference in the identity of the nucleophilic species would lead one to predict that the chain elongation reaction has more S N 1-like character than the protein prenylation reaction. A thiolate is a very powerful nucleophile, and thus is able to push the pyrophosphate leaving group off, implying some degree of S N 2 character. The electrons in a pi bond, in contrast, are only weakly nucleophilic, and thus need to be pulled in by a powerful electrophile - ie . a carbocation.
So it makes perfect sense that the chain elongation reaction should more S N 1-like than S N 2-like. Is this in fact the case? We know how to answer this question experimentally - just run the reaction with fluorinated DMAPP or GPP substrates and observe how much the fluorines slow things down.
If the reaction is S N 1-like, the electron-withdrawing fluorines should destabilize the allylic carbocation intermediate and thus slow the reaction down considerably. If the mechanism is S N 2-like, the fluorine substitutions should not have a noticeable effect, because a carbocation intermediate would not be formed. When this experiment was performed with FPP synthase, the results were dramatic: the presence of a single fluorine slowed down the rate of the reaction by a factor of about 60, while two and three fluorines resulted in a reaction that was 500,000 and 3 million times slower, respectively ( J. Am. Chem. Soc . 1981 , 103 , 3926.) These results strongly suggest indicate the formation of a carbocation intermediate in an S N 1-like displacement.
In this section, we will briefly examine the reaction catalyzed by an enzyme called squalene synthase, an important enzymatic transformation that involves some very interesting and unusual electrophilic additions, rearrangements, and reactive intermediates. This particular enzyme is also of interest because it represents a potential new target for cholesterol-lowering drugs.
Cholesterol, as we discussed earlier in this chapter, is derived from a 30-carbon isoprenoid molecule called squalene. Squalene, in turn, is derived from the condensation of two molecules of farnesyl diphosphate (FPP), a 15-carbon isoprenoid. You may recall that FPP is the product of the C 4 to C 1 , or 'head to tail' electrophilic condensation of isoprenoid chains:
The condensation of two molecules of FPP to form squalene, however, is something different: this is a 'head to head' condensation, where C 1 of the first molecule forms a bond to C 1 of the second. The chemistry involved is quite a bit more complicated.
The first two steps are familiar: first, the pyrophosphate on one FPP molecule leaves (step 1), resulting in an allylic carbocation that is attacked by the C 2 -C 3 π bond of the second molecule (step 2).
This results in a new carbon-carbon bond between the two FPP molecules, but with incorrect C 1 to C 2 connectivity (remember, the overall reaction is a C 1 to C 1 condensation). In step 3, a proton is abstracted and the electrons from the broken C-H bond bridge across a 2-carbon gap to form a cyclopropyl intermediate.
In the second stage of squalene synthesis, the second pyrophosphate group leaves, generating a cyclopropylcarbinyl cation (step 4). Because this is a primary carbocation, you probably are wondering about how stable it could be (and thus how likely an intermediate). As it turns out, such carbocations are remarkably stable, due to favorable interactions between the empty orbital and orbitals on the three-membered ring (the level of bonding theory needed to really understand this idea is beyond the scope of this text, but you may learn about it if you take a class in advanced organic chemistry). What occurs next is an alkyl shift leading to a tertiary carbocation (step 5).
Discussion of the final step (step 6) will need to be put off - this is a reduction with a hydride nucleophile derived from a coenzyme called NADPH. Although this may seem like an extremely convoluted (and perhaps unlikely!) mechanism, there is much experimental evidence to back it up.
Exercise \(\PageIndex{1}\)
Farnesyl diphosphate (FPP) is synthesized by adding another five-carbon building block to geranyl diphosphate. What is this building block - IPP or DMAPP? Draw a mechanism for the formation of FPP.
- Answer
-
IPP is the five-carbon species added to geranyl diphosphate to make the 15-
carbon farnesyl diphosphate.
Exercise \(\PageIndex{2}\)
Propose a likely mechanism for the following transformation, which is the first stage in a somewhat complex reaction in the synthesis of an isoprenoid compound in plants. (Science 1997, 277, 1815)
- Answer
-
Exercise \(\PageIndex{3}\)
Propose a likely mechanism for the following transformation.
- Answers
-


3.9: Summary
3.1 The Laws of Planetary Motion
Tycho Brahe’s accurate observations of planetary positions provided the data used by Johannes Kepler to derive his three fundamental laws of planetary motion. Kepler’s laws describe the behavior of planets in their orbits as follows: (1) planetary orbits are ellipses with the Sun at one focus; (2) in equal intervals, a planet’s orbit sweeps out equal areas; and (3) the relationship between the orbital period ( P ) and the semimajor axis ( a ) of an orbit is given by P 2 = a 3 (when a is in units of AU and P is in units of Earth years).
3.2 Newton’s Great Synthesis
In his Principia , Isaac Newton established the three laws that govern the motion of objects: (1) objects continue to be at rest or move with a constant velocity unless acted upon by an outside force; (2) an outside force causes an acceleration (and changes the momentum) for an object; and (3) for every action there is an equal and opposite reaction. Momentum is a measure of the motion of an object and depends on both its mass and its velocity. Angular momentum is a measure of the motion of a spinning or revolving object and depends on its mass, velocity, and distance from the point around which it revolves. The density of an object is its mass divided by its volume.
3.3 Newton’s Universal Law of Gravitation
Gravity, the attractive force between all masses, is what keeps the planets in orbit. Newton’s universal law of gravitation relates the gravitational force to mass and distance:
The force of gravity is what gives us our sense of weight. Unlike mass, which is constant, weight can vary depending on the force of gravity (or acceleration) you feel. When Kepler’s laws are reexamined in the light of Newton’s gravitational law, it becomes clear that the masses of both objects are important for the third law, which becomes a 3 = ( M 1 + M 2 ) × P 2 . Mutual gravitational effects permit us to calculate the masses of astronomical objects, from comets to galaxies.
3.4 Orbits in the Solar System
The closest point in a satellite orbit around Earth is its perigee, and the farthest point is its apogee (corresponding to perihelion and aphelion for an orbit around the Sun). The planets follow orbits around the Sun that are nearly circular and in the same plane. Most asteroids are found between Mars and Jupiter in the asteroid belt, whereas comets generally follow orbits of high eccentricity.
3.5 Motions of Satellites and Spacecraft
The orbit of an artificial satellite depends on the circumstances of its launch. The circular satellite velocity needed to orbit Earth’s surface is 8 kilometers per second, and the escape speed from our planet is 11 kilometers per second. There are many possible interplanetary trajectories, including those that use gravity-assisted flybys of one object to redirect the spacecraft toward its next target.
3.6 Gravity with More Than Two Bodies
Calculating the gravitational interaction of more than two objects is complicated and requires large computers. If one object (like the Sun in our solar system) dominates gravitationally, it is possible to calculate the effects of a second object in terms of small perturbations. This approach was used by John Couch Adams and Urbain Le Verrier to predict the position of Neptune from its perturbations of the orbit of Uranus and thus discover a new planet mathematically.


3.5: Counting Methods
Recall that
\[P(A)=\frac{\text { number of ways for } A \text { to occur }}{\text { total number of outcomes }} \nonumber \]
for theoretical probabilities. So far the problems we have looked at had rather small total number of outcomes. We could easily count the number of elements in the sample space. If there are a large number of elements in the sample space we can use counting techniques such as permutations or combinations to count them.
Multiplication Principle and Tree Diagrams
The simplest of the counting techniques is the multiplication principle. A tree diagram is a useful tool for visualizing the multiplication principle.
Let’s say that a person walks into a restaurant for a three course dinner. There are four different salads, three different entrees, and two different desserts to choose from. Assuming the person wants to eat a salad, an entrée and a desert, how many different meals are possible?
Solution
Looking at the tree diagram we can see that the total number of meals is \(4 \times 3 \times 2 = 24\, \text{meals}\).
If there are \(n_1\) ways to of choosing the first item, \(n_2\) ways of choosing the second item after the first item is chosen, \(n_3\) ways of choosing the third item after the first two have been chosen, and so on until there are \(n_k\) ways of choosing the last item after the earlier choices, then the total number of choices overall is given by
\[n_1 \times n_2 \times n_3 \times n_4 \times n_5 ... \times n_k. \label{Multiplication Principle} \]
Let’s look at the number of ways that four people can line up. We can choose any of the four people to be first. Then there are three people who can be second and two people who can be third. At this point there is only one person left to be last. Using the multiplication principle there are
\[4 \times 3 \times 2 \times 1 = 24\, \text{ways} \nonumber \]
for four people to line up.
This type of calculation occurs frequently in counting problems so we have some notation to simplify the problem.
The factorial of \(n\), read “n factorial” is
\[n! = n(n-1)(n-2)(n-3)...(2)(1). \nonumber \]
By this definition, \(0! = 1.\)
\[\begin{align*} 5! &= 5 \times 4 \times 3 \times 2 \times 1 \\[4pt] &= 120 \\[5pt] 8! &= 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \\[4pt] &= 40,320 \end{align*} \nonumber \]
Factorials get very large very fast.
\[20! = 2.43 \times 10^{18} \nonumber \]
and
\[40! = 8.16 \times 10^{47}. \nonumber \]
70! is larger than most calculators can handle.
The multiplication principle may seem like a very simple idea but it is very powerful. Many complex counting problems can be solved using the multiplication principle.
Some license plates in Arizona consist of three digits followed by three letters. How many license plates of this type are possible if:
- There are 10 digits (0, 1, 2, 3, …, 9) and 26 letters. \[\underbrace{(10 \cdot 10 \cdot 10) \cdot (26 \cdot 26 \cdot 26)}_{\text{digits}}=17,576,000 \text {license plates } \nonumber \]
- letters can be repeated but digits cannot? \[\underbrace{(\underline{10} \cdot \underline{9} \cdot \underline{8}) \cdot(26 \cdot 26 \cdot 26)}_{\text {letters}}=12,654,720\, \text{license plates} \nonumber \]
- the first digit cannot be zero and both digits and letters can be repeated? \[\underbrace{(\underline{9} \cdot \underline{10} \cdot \underline{10})}_{\text{digits}} \cdot \underbrace{(\underline{26} \cdot 26 \cdot 26)}_{\text{letters}}=15,818,400\, \text{license plates} \nonumber \]
- neither digits nor numbers can be repeated. \[\underbrace{(\underline{10} \cdot \underline{9} \cdot \underline{8})}_{\text{digits}} \cdot \underbrace{(\underline{26} \cdot \underline{25} \cdot \underline{24})}_{\text{letters}}=11,232,000 \, \text{license plates} \nonumber \]
Permutations
Consider the following counting problems:
- In how many ways can three runners finish a race?
- In how many ways can a group of three people be chosen to work on a project?
What is the difference between these two problems? In the first problem the order that the runners finish the race matters. In the second problem the order in which the three people are chosen is not important, only which three people are chosen matters.
A permutation is an arrangement of a set of items. The number of permutations of n items taking r at a time is given by:
\[P(n, r)=\frac{n !}{(n-r) !} \label{permutation} \]
Note: Many calculators can calculate permutations directly. Look for a function that looks like \(_nP_r\) or \(P(n,r)\)
Let’s look at a simple example to understand the formula for the number of permutations of a set of objects. Assume that 10 cars are in a race. In how many ways can three cars finish in first, second and third place? The order in which the cars finish is important. Use the multiplication principle. There are 10 possible cars to finish first. Once a car has finished first, there are nine cars to finish second. After the second car is finished, any of the eight remaining cars can finish third. 10 x 9 x 8 = 720. This is a permutation of 10 items taking three at a time.
Using the permutation formula (Equation \ref{permutation}):
\[P(10,3)=\frac{10 !}{(10-3) !}=\frac{10 !}{7 !}= \dfrac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 10 \cdot 9 \cdot 8 = 720 \nonumber \]
Using the multiplication principle (Equation \ref{Multiplication Principle}):
\[\underline{10} \cdot \underline{9} \cdot \underline{8} = 720 \nonumber \]
There are 720 different ways for cars to finish in the top three places.
The school orchestra is planning to play six pieces of music at their next concert. How many different programs are possible?
Solution
This is a permutation because they are arranging the songs in order to make the program. Using the permutation formula (Equation \ref{permutation}):
\[P(6,6)=\frac{6 !}{(6-6) !}=\frac{6 !}{0 !}=\frac{720}{1}=720 \nonumber \]
Using the multiplication principle (Equation \ref{Multiplication Principle}):
\[\underline{6} \cdot \underline{5} \cdot \underline{4} \cdot \underline{3} \cdot \underline{2} \cdot \underline{1}=720 \nonumber \]
There are 720 different ways of arranging the songs to make the program.
The Volunteer Club has 18 members. An election is held to choose a president, vice-president and secretary. In how many ways can the three officers be chosen?
Solution
The order in which the officers are chosen matters so this is a permutation.
Using the permutation formula (Equation \ref{permutation}):
\[P(18,3)=\frac{18 !}{(18-3) !}=\frac{18 !}{15 !}=18 \cdot 7 \cdot 6=4896 \nonumber \]
Note: All digits in 18! in the numerator from 15 down to one will cancel with the 15! in the denominator.
Using the multiplication principle (Equation \ref{Multiplication Principle}):
\(\begin{array} {cccccc} {\underline{18}}&{\cdot}&{\underline{17}}&{\cdot}&{\underline{16}}&{ = 4896}\\ {\text{Pres.}}&{}&{\text{V.P.}}&{}&{\text{Sec.}}&{} \end{array}\)
There are 4896 different ways the three officers can be chosen.
Another notation for permutations is \(_nP_r\). So, \(P(18,3)\) can also be written as \(_{18}P_3\). Most scientific calculators have an \(_nP_r\) button or function.
Combinations
Choose a committee of two people from persons A, B, C, D and E. By the multiplication principle there are \*5 \cdot 4 = 20\) ways to arrange the two people.
AB AC AD AE BA BC BD BE CA CB
CD CE DA DB DC DE EA EB EC ED
Committees AB and BA are the same committee. Similarly for committees CD and DC. Every committee is counted twice.
\[\frac{20}{2}=10 \nonumber \]
so there are 10 possible different committees.
Now choose a committee of three people from persons A, B, C, D and E. There are \(5 \cdot 4 \cdot 3 = 60\) ways to pick three people in order. Think about the committees with persons A, B and C. There are \(3! =6\) of them.
ABC ACB BAC BCA CAB CBA
Each of these is counted as one of the 60 possibilities but they are the same committee. Each committee is counted six times so there are
\[\frac{60}{6}=10 \, \text{different committees}. \nonumber \]
In both cases we divided the number of permutations by the number of ways to rearrange the people chosen.
The number of permutations of n people taking r at a time is \(P(n,r)\) and the number of ways to rearrange the people chosen is \(r!\). Putting these together we get
\[\begin{aligned}
\frac{n !}{\# \text { ways to arrange r items }} &=\frac{P(n, r)}{r !}=\frac{(n-r) !}{\frac{r !}{1}} \\
&=\frac{n !}{(n-r) !} \cdot \frac{1}{r !} \\
&=\frac{n !}{(n-r) ! r !}
\end{aligned} \nonumber \]
A combination is a selection of objects in which the order of selection does not matter. The number of combinations of n items taking r at a time is:
\[C(n, r)=\frac{n !}{r !(n-r) !} \label{combination} \]
Note: Many calculators can calculate combinations directly. Look for a function that looks like \(_nC_r\) or (C(n,r)\) .
A student has a summer reading list of eight books. The student must read five of the books before the end of the summer. In how many ways can the student read five of the eight books?
Solution
The order of the books is not important, only which books are read. This is a combination of eight items taking five at a time.
\[C(8,5)=\frac{8 !}{5 !(8-5) !}=\frac{8 !}{5 ! 3 !}= \dfrac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1} = \dfrac{8 \times 7 \times 6}{3 \times 2 \times 1} = 8 \times 7 = 56 \nonumber \]
There are 56 ways to choose five of the books to read.
A child wants to pick three pieces of Halloween candy to take in her school lunch box. If there are 13 pieces of candy to choose from, how many ways can she pick the three pieces?
Solution
This is a combination because it does not matter in what order the candy is chosen.
\[\begin{align*} C(13,3) &=\frac{13 !}{3 !(13-3) !}=\frac{13 !}{3 ! 10 !} \\[4pt] &= \dfrac{13 \times 12 \times 11 \times 10 \times 9\times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1)(10 \times 9\times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \\[4pt] &=\frac{13 \times 12 \times 11}{3 \times 2 \times 1} \\[4pt] =\frac{1716}{6}=286 \end{align*} \nonumber \]
There are 286 ways to choose the three pieces of candy to pack in her lunch.
Note: The difference between a combination and a permutation is whether order matters or not. If the order of the items is important, use a permutation. If the order of the items is not important, use a combination.
A serial number for a particular model of bicycle consists of a letter followed by four digits and ends with two letters. Neither letters nor numbers can be repeated. How many different serial numbers are possible?
Solution
This is a permutation because the order matters.
Use the multiplication principle to solve this. There are 26 letters and 10 digits possible.
\[26 \cdot 10 \cdot \underline{9} \cdot \underline{8} \cdot \underline{7} \cdot \underline{25} \cdot \underline{24}=78,624,000 \nonumber \]
There are 78,624,000 different serial numbers of this form.
A class consists of 15 men and 12 women. In how many ways can two men and two women be chosen to participate in an in-class activity?
Solution
This is a combination since the order in which the people is chosen is not important.
Choose two men:
\[C(15,2)=\frac{15 !}{2 !(15-2) !}=\frac{15 !}{2 ! 13 !}=105 \nonumber \]
Choose two women:
\[C(12,2)=\frac{12 !}{2 !(12-2) !}=\frac{12 !}{2 ! 10 !}=66 \nonumber \]
We want 2 men and 2 women so multiply these results.
\[105(66)=6930 \nonumber \]
There are 6930 ways to choose two men and two women to participate.
Probabilities Involving Permutations and Combinations
Now that we can calculate the number of permutations or combinations, we can use that information to calculate probabilities.
There are 20 students in a class. Twelve of the students are women. The names of the students are put into a hat and five names are drawn. What is the probability that all of the chosen students are women?
Solution
This is a combination because the order of choosing the students is not important.
\[P(\text { all females })=\frac{\# \text { ways to pick } 5 \text { women }}{\# \text { ways to pick } 5 \text { students }} \nonumber \]
The number of way to choose 5 women is
\[C(12,5)=792 \nonumber \]
The number of ways to choose 5 students is
\[C(20,5)=15,504 \nonumber \]
\[P(\text { all females })=\frac{\# \text { ways to pick } 5 \text { women }}{\# \text { ways to pick } 5 \text { students }}=\frac{792}{15,504}=0.051 \nonumber \]
The probability that all the chosen students are women is 0.051 or 5.1%.
The local Boys and Girls Club holds a duck race to raise money. Community members buy a rubber duck marked with a numeral between 1 and 30, inclusive. The box of 30 ducks is emptied into a creek and allowed to float downstream to the finish line. What is the probability that ducks numbered 5, 18 and 21 finish in first, second and third, respectively?
Solution
This is a permutation since the order of finish is important.
\[P(5,18 \& 21 \text { finish } 1 \text { st, } 2 \text { nd } \& \text { 3 } \mathrm{rd})=\dfrac{\# \text { ways } 5,18 \& 21 \text { finish } 1 \text { st, } 2 \text { nd } \& 3 \mathrm{rd}}{\# \text { ways any ducks can finish } 1 \text { st, } 2 \text { nd } \& \text { 3rd }} \nonumber \]
There is only one way that the ducks can finish with #5 in first, #18 in second and #21 in third.
The number of ways any ducks can finish in first, second and third is
\[P(30,3)=24,360 \nonumber \]
\[ \begin{align*} P(5,18 \& 21 \text { finish } 1 \text { st, } 2 \text { nd } \& \text {3rd}) &=\frac{\# \text { ways } 5,18 \& 21 \text { finish } 1 \text { st, } 2 \text { nd } \& 3 \text { rd }}{\# \text { ways any ducks can finish } 1 \text {st,}, 2 \text {nd} \& \text {3 rd}} \\[4pt] &=\frac{1}{24,360} \approx 4.10 x 10^{-5} \end{align*} \nonumber \]
The probability that ducks numbered 5, 18 and 21 finish in first, second and third, respectively, is approximately 0.000041 or 0.0041%.
A poker hand consists of two cards. What is the probability that the poker hand consists of two jacks or two fives?
Solution
It’s not possible to get two jacks and two fives at the same time so these are mutually exclusive events.
The number of ways to get two jacks is
\[C(4,2)=6. \nonumber \]
The number of ways to get two fives is
\[C(4,2)=6 \nonumber \]
The number of ways to get two jacks or two fives is
\[6+6=12 \nonumber \]
The total number of ways to get a 2-card poker hand is
\[C(52,2)=1326 \nonumber \]
\[P(2 \text { jacks or } 2 \text { fives })=\dfrac{\text { number of ways to get } 2 \text { jacks or } 2 \text { fives }}{\text { number of ways to choose } 2 \text { cards }}=\frac{12}{1326} \approx 0.009 \nonumber \]
The probability of getting two jacks or two fives is approximately 0.009 or 0.9%.
A basket contains 10 good apples and two bad apples. If a distracted shopper reaches into the basket and picks three apples without looking, what is the probability he gets one bad apple?
Solution
This is a combination since the order in which the apples were picked is not important. He picks three apples total. If one apple is bad the other two must be good. Find the probability of one bad apple and two good apples.
\[P(\text { one bad and two good apples })=\frac{\# \text { ways to get one bad and two good apples }}{\# \text { ways to get three apples }} \nonumber \]
The number of ways to get one bad and two good apples is
\[C(2,1) \cdot C(10,2)=2 \cdot 45=90 \nonumber \]
The number of ways to get three apples is
\[C(10,3)=120 \nonumber \]
\[ \begin{align*} P(\text { one bad and two good apples }) &=\frac{\# \text { ways to get one bad and two good apples }}{\# \text { ways to get three apples }} \\[4pt] &=\frac{90}{120} \\[4pt] &=0.75 \end{align*} \nonumber \]
The probability of getting one bad apple out of three apples is 0.75 or 75%.2