<s>[INST] An Urn with 60 marbles A urn contains 60 marbles 10 red 10 white 10 blue 10 green 10 brown and 10 orange. I draw from the urn 6 times. What is the probability I have exactly 5 varieties?
Here's what I've got,
There are $10^5$ ways I can draw 5 different marbles. So the number of ways I am unable to draw the 6th variety should be $9 \cdot 5$ ways? Thus $\dfrac{10^5 \cdot 45}{C(60,10)}$ is the probability of having exactly 5 varietys?
 [/INST] Think of the marbles as distinct (write an ID number on each). We assume the drawing is done without replacement. 
There are $\binom{60}{6}$ equally likely ways to choose $6$ marbles.
Now we count the "favourables." If there are $5$ types, we must have $2$ of one type, and $1$ each of $4$ other types. 
There are $\binom{6}{1}$ ways to choose the colour we will have $2$ in. For each of these ways, there are $\binom{10}{2}$ ways of choosing the $2$ marbles. For each of these ways, there are $\binom{5}{4}$ ways to choose the other $4$ colours that will be represented. And for each such choice, there are $10^4$ ways to choose the actual marbles, for a probability of 
$$\frac{\binom{6}{1}\binom{10}{2}\binom{5}{4}10^4}{\binom{60}{6}}.$$
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