There are members in the U.S. Senate ( from each state). In how many ways can a committee of senators be formed if no state may be represented more than once?
Source: NCTM Mathematics Teacher, August 2006
We want to organize the senators into -member committees with each state sending only one senator to the committees.
Ways to choose states from states =
Ways to send one of two senators from each of the five states =
Ways to form committees =
We select senators from a group of to form -member committees. First, we pick one senator from ; second, we pick one from the remaining (not because we cannot have senators from the same state); third, we pick one from , etc. Hence the number of ways to pick senators = . Each way is an ordering (permutation) of the senators and the number of orderings of the objects in a set of objects equals . Since the order of selection is not important, we divide the number of ways by 5! to get