At least one department unrepresented on a committee

Question

A company has three departments, each with exactly $4$ employees ( $12$ in all). A committee of $4$ is chosen at random.

What is the probability that at least one department is left with nobody on the committee?

Accepted Answer

Let Ad be the event that department d is unrepresented, and apply inclusion-exclusion over the three departments. Counts are over \binom{12}{4} = 495 equally likely committees. - One department absent: the four members come from the other 8, so \binom{8}{4} = 70, with \binom{3}{1} = 3 departments. - Two departments absent: all four come from one department of 4, so \binom{4}{4} = 1, with \binom{3}{2} = 3 pairs. - All three departments absent is impossible. Alternating signs: 3(70) - 3(1) = 210 - 3 = 207. So P(	ext{some department absent}) = \frac{207}{495} = \frac{23}{55} \approx 0.4182. Tak…

At least one department unrepresented on a committee

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