Seven dinners
Task number: 3493
Peter Generous hosts a dinner for his friends every evening. Three guests are invited to each dinner. In how many ways can Peter send invitations to his 7 friends for an entire week such that each of these seven friends is invited at least once?
Solution
For each dinner Peter may invite one of \(\binom73\) possible triples, and so for the entire week this gives \(\binom73^7\) possibilities, though this includes possibilities in which some guest is never invited. If one particular individual is never invited, we can follow the same reasoning with six hosts, so there are \(\binom63^7\) possibilities. If two friends are never invited, there are \(\binom53^7\) possibilities, however we can choose this pair in \(\binom72\) ways. We continue similarly for larger numbers of univited friends.
We now compute the numerical result: \( \binom{7}{3}^7 - 7 \binom{6}{3}^7 + 21 \binom{5}{3}^7 - 35 \binom{4}{3}^7 + 35 \binom{3}{3}^7=35^7-7{\cdot} 20^7+21{\cdot}10^7-35{\cdot}4^7+35\\ =55{,}588,723{,}470 \)
Answer
There are a total of \( \binom{7}{3}^7 - 7 \binom{6}{3}^7 + 21 \binom{5}{3}^7 - 35 \binom{4}{3}^7 + 35 \binom{3}{3}^7=55{,}588,723{,}470 \) possible ways to arrange the invitations.
