+471 I have \(n\) friends. Every night of the 365-day year I invite 4 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 four friends?
+349 The minimum is 7. 6 choose 4 is 360, which is a bit too low, and 8 choose 4 is 1680, but 7 choose 4 is 840, which is the minimum amount of friends you need.
But seriously, why do you want the minimum amount of friends? Don't you want to have the most friends possible? 
+129852 Here's an graphical way of figuring this out
C (n , 4) = n! / ( [ n - 4] ! * 4! ) .....so....we require that.......
C ( n , 4 ) ≥ 365
n! / ( [ n - 4] ! * 4! ) ≥ 365
n! / [ n - 4]! ≥ 4! * 365
n! / [ n - 4]! ≥ 8760
n * (n - 1) (n - 2) (n - 3) ≥ 8760
Look at the graph here :
https://www.desmos.com/calculator/j9rafrlbpn
Note that the minimum n that makes this true ≈ 11.3
So.....we need 12 friends
Proof C(11, 4) = 330 too few groups of 4
But C( 12. 4) = 495 .......we even have a few "leftover" groups of 4 !!!!!!
