I have n friends. Every night of the 365-day year I invite three of them to dinner. What is the smallest n could be such that it is still possible for me to make these invitations without ever inviting the same group of three friends? (I can invite the same friend, or even the same pair of friends, but never the same whole group of three.)
We need to find this
C ( n , 3) = 365
__________ = 365
(n - 3)! 3!
(n) ( n - 1) (n - 2) = 365 * 3!
n (n^2 - 3n + 2) = 2190
n^3 - 3n^2 + 2n - 2190 = 0
Using a little technology, n ≈ 14.012
So....you would need 15 friends
P.S. ....if there were only 364 nights you could invite exactly 14 friends !!!