<s>[INST] What is the probability of the office being deemed functional? An IT office consists of 10 computers of which exactly 4 are working. To check if the office is functional, officials inspect four of the computers picked at random (without replacement). The office is deemed functional if at least three of the four computers are working. What is the probability of the office being deemed functional?
My Solution:
$E$ be event of office being deemed functional.
$E = 3$ out of $4$ computers working or all four computers working
$P(E) = P(3~\text{out of}~4~\text{computers working}) + P(\text{all four computers working})$
(3 out of 4 computers working)={WWWD,WWDW,WDWW,DWWW} where W,D are Working, Defective computers respectively.
(all four computers working)={WWWW}
$P(3~\text{out of}~4~\text{computers working}) = \frac{4 \cdot 3 \cdot 2 \cdot 6}{10 \cdot 9 \cdot 8 \cdot 7} \cdot 4 = 0.114$
$P(\text{all four computers working}) = \frac{4 \cdot 3 \cdot 2 \cdot 1}{10 \cdot 9 \cdot 8 \cdot 7} = 0.004$
$P(E) = 0.114+0.004 = 0.118$
Solution Provided is:
$P(3~\text{out of}~\text{4}~\text{computers working}) = \frac{{4 \choose 3}{6 \choose 1}}{{10 \choose 4}}$
$P(\text{all four computers working}) = \frac{{4 \choose 4}{6 \choose 0}}{{10 \choose 4}}$
The sum of these approximately equals my solution. Looking at the formula though I feel it should not work. The denominator ${10 \choose 4}$ is not size of the sample space for given problem, since all $10$ elements are not distinct. Also ${4 \choose 3}{6 \choose 1} = 24$ is not the number of possibilities that satisfy the condition ($3$ out of $4$ computers working), only $4$ possibilities satisfy this. Why does this answer still work?
 [/INST] The two answers are numerically exactly the same.  Yours is $\frac{576}{5040}+\frac{24}{5040}=\frac{600}{5040}$ and the provided solution is $\frac{24}{210}+\frac{1}{210}=\frac{25}{210}$.  Both are $\frac{5}{42} \approx 0.1190476$.
The factor of $4!=24$ in the numerator and denominator is simply the number of ways of ordering the four selected computers. You distinguished the orderings while the provided solution did not.
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