Problem:
A fancy bed and breakfast inn has 5 rooms, each with a distinctive color-coded decor. One day 5 friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than 2 friends per room. In how many ways can the innkeeper assign the guests to the rooms?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
If each friend rooms alone, then there are ways to assign the guests to the rooms. If one pair of friends room together and the others room alone, then there are ways to choose the roommates and then ways to assign the rooms to the 4 sets of occupants, for a total of possible arrangements. The only other possibility is to have two sets of roommates. In this case the roommates can be chosen in ways (choose the solo lodger first), and then there are ways to assign the rooms, for a total of possibilities. Therefore the answer is .
The problems on this page are the property of the MAA's American Mathematics Competitions