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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?
 

 Jun 12, 2020
edited by Guest  Jun 12, 2020
 #1
avatar+1130 
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n=12 since \(365\)<\(\dbinom n4\) and \(\dbinom {11}4\)=330 and \(\dbinom {12}4\)=495 so n=12

 Jun 12, 2020
 #2
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Thank you for trying, buy that is not right :/

 Jun 12, 2020
 #3
avatar+48 
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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)  laugh

 Jun 20, 2020
edited by trumpstinks  Jun 20, 2020

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