I run a book club with $n$ people, not including myself. Every day, for $100$ days, I invite $14$ members in the club to review a book. What is the smallest positive integer $n$ so that I can avoid ever having the exact same group of $14$ members over all $100$ days?
There are ${n \choose 14}$ ways to choose the $14$ members, so we must have ${n \choose 14} \ge 100$. Checking small $n$, we find ${10 \choose 14} = 0$, ${11\choose 14} = 0$, ${12 \choose 14} = 0$, ${13 \choose 14} = 0$, but ${14 \choose 14} = 1$, so $n = \boxed{14}$ is the least value that satisfies this inequality.
