4. $[\mathbf{2 0}]$ Thomas and Michael are just two people in a large pool of well qualified candidates for appointment to a problem writing committee for a prestigious college math contest. It is 40 times more likely that both will serve if the size of the committee is increased from its traditional 3 members to a whopping $n$ members. Determine $n$. (Each person in the pool is equally likely to be chosen.)
Answer: 16. Suppose there are $k$ candidates. Then the probability that both serve on a 3 membered committee is $(k-2) /\binom{k}{3}$, and the odds that both serve on an $n$ membered committee are $\binom{k-2}{n-2} /\binom{k}{n}$. The ratio of the latter to the former is
$$
\frac{\binom{k}{3}\binom{k-2}{n-2}}{(k-2)\binom{k}{n}}=\frac{k!(k-2)!1!(k-3)!n!(k-n)!}{k!(k-2)!(n-2)!(k-n)!3!(k-3)!}=\frac{n \cdot(n-1)}{3!} .
$$

Solving $n \cdot(n-1)=240$ produces $n=16,-15$, and we discard the latter.