I run a book club with n people, not including myself. Every day, for 365 days, I invite five 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 five members over all 356 days?
+1262 n=12 since \(365\)<\(\dbinom n4\) and \(\dbinom {11}4\)=330 and \(\dbinom {12}4\)=495 so n=12
+51 jimkey17, you had the right idea, but you made a mistake in your work. It should be choose 5, not 4. The real answer is 11, as 11 choose 4 > 365, but 10 choose 4 isn't. (Note: I solved this problem on AoPS, and 11 is correct, so you need have no doubts in the solution) 
