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 $365$ days?

11 members.

Step-by-step explanation:

When we have a number N of objects and we want to divide it into groups K objects, the number of possible combinations is:

Here we have that C must be at least, 365 and K = 5, we want to find the value of N.

Now, you can play with different numbers of N, for example, if we take N = 9 we have:

C = 9!/(4!*5!) = 9*8*7*6/(4*3*2) = 126

so 9 is not enough, let's see N = 10

C = 10!/(5!*5!) = 10*9*8*7*6/(5*4*3*2) = 252

10 is not enough, let's see with 11.

C = 11!/(6!*5!) = 11*10*9*8*7/(5*4*3*2) = 462

So you need at least 11 members in the club.

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