Suppose a person has 7 friends, whom he wants to invite for dinner over a week. Here are the rules:
- Each night, exactly 3 people will be invited.
- Any pair of friends is invited on at most one common night. Namely for any $i \neq j$, $i$ and $j$ can both be invited only on at most one night.
How many possible ways are there to achieve this task? Any help is appreciated.
[I've tried to argue via subsets of a set. Here is an interpretation. Take any 7-element set $A$, we want to count the number of 7-element sets $B$ where for each $B_i\in B$, $B_i$ is a 3-element subset of $A$, and for any $i\neq j$, $|B_i\cap B_j|\leq 1$. Once we have all such sets, we basically need to multiply the total number of all such $B$'s by $7!$].