Question

A 20-person fraternity is making their weekend plans around 5 different parties in IV. Each brother...

A 20-person fraternity is making their weekend plans around 5 different parties in IV. Each brother will attend exactly one of the parties. Fraternity rules state that the fraternity must have a representative at each party. Assume that only the number of brothers that show up at a party is important, i.e. the brothers are indistinguishable.

(a) How many different arrangements of plans are possible?

20-15 brothers are free to attend any of the 5 parties

(b) How many different arrangements of plans are possible so that none of the bros in the fraternity have to attend a party alone?

(c) Suppose one specific party is planning to have pizza bites and so the five most senior bros insist on going to that one. Now how many different arrangements of plans are possible?

(d) How many different arrangements are possible if the rule is relaxed and there may be parties with no fraternity members?