I run a book club with n people, not including myself. Every day, for 100 days, I invite 4 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 4 members over all 100 days?
We are looking for the smallest n such that n(n - 1)(n - 2)(n - 3) > 100.
Since 4*3*2*1 = 24 and 5*4*3*2 = 120, the smallest n that works is n = 5.
For each day, we have 4 distinct members to pick from. Hence, the number of possible combinations of 4 members is given by n choose 4, or nC4. The smallest positive integer n such that nC4 is greater than or equal to 100 is n = 23. This means that with a club of at least 23 members, it's possible to pick 4 different members every day for 100 days without ever having the exact same group twice.
You have the right idea, but how did you get n=23
23 C 4 = 8,855 !! All you need is that: nC4 >=100 days
So, n = 9, because: 9 C 4 = 126 unique combinations - so that 4 DISTINCT people can easily be picked out of 9 for at least 100 days without having the exact same group of 4 people reviewing a book.