AMC 12 2014a Problem 13

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?

$\textbf{(A) }2100 \textbf{(B) }2220 \textbf{(C) }3000 \textbf{(D) }3120 \textbf{(E) }3125$