I have n friends Every night of the 365 day year I invite fo

I have n friends. Every night of the 365 day year I invite four of them to dinner. What is the smallest n there could be such that it is still possible for me to make these invitations without ever inviting the same group of four friends? (I can invite the same friend more than once, or the same pair of friends more than once, or the same group of three friends more than once, but NOT the same group of four friends more than once).

Solution

Daily night one set of 4 persons for the dinner

The condition is same group not repeated for any other day

This is total no of ways - no of ways of inviting the same group for all 365 days

If there are n friends nC4 should be selected

next day minimum one person should be changed hence n-1C4

n-2 C4 and sio on.

Thus minimum n should be such that

n-365C4 is valid

n =369

If n>=369 this is possible.