I run a book club with $n$ people, not including myself. Every day, for $365$ days, I invite four 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 four members over all $365$ days?

12

Step-by-step explanation:

If we have n people and we want to do groups of 4, the total number of different combinations is:

and we want find the smallest n such c > 365.

let's do it by brute force:

if n = 5, we have:

if n = 6

if n = 7

if n = 8

if n = 9

if n = 10

So you can see the patern here.

So we have the previous number multiplied by n and divided by n - 4

if n = 11

C = 210*11/(11- 4) = 330

if n = 12

C = 330*12/(12 - 4) = 495

Then the minimum number of members such we have more than 365 combinations is 12 members

c. (3,2,6) is the answer.