MathDB
Fancy Bed and Breakfast

Source: 2014 AMC 12A #13

February 5, 2014
countingdistinguishabilitybinomial coefficientscomplementary countingAMC

Problem Statement

A fancy bed and breakfast inn has 55 rooms, each with a distinctive color-coded decor. One day 55 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 22 friends per room. In how many ways can the innkeeper assign the guests to the rooms?
<spanclass=latexbold>(A)</span>2100<spanclass=latexbold>(B)</span>2220<spanclass=latexbold>(C)</span>3000<spanclass=latexbold>(D)</span>3120<spanclass=latexbold>(E)</span>3125<span class='latex-bold'>(A) </span>2100\qquad <span class='latex-bold'>(B) </span>2220\qquad <span class='latex-bold'>(C) </span>3000\qquad <span class='latex-bold'>(D) </span>3120\qquad <span class='latex-bold'>(E) </span>3125\qquad