## U.S. Senate Committee

There are $100$ members in the U.S. Senate ($2$ from each state). In how many ways can a committee of $5$ senators be formed if no state may be represented more than once?
Source: NCTM Mathematics Teacher, August 2006

Solution
We want to organize the $100$ senators into $5$-member committees with each state sending only one senator to the committees.
Ways to choose $5$ states from $50$ states = $\dbinom{50}{5}$
Ways to send one of two senators from each of the five states = $2^5$
Ways to form committees = $\dbinom{50}{5}\times 2^5=67,\!800,\!320$

Answer: $67,\!800,\!320$

Alternative solution
We select $5$ senators from a group of $100$ to form $5$-member committees. First, we pick one senator from $100$; second, we pick one from the remaining $98$ (not $99$ because we cannot have $2$ senators from the same state); third, we pick one from $96$, etc. Hence the number of ways to pick $5$ senators = $100\times 98\times 96\times 94\times 92$. Each way is an ordering (permutation) of the senators and the number of orderings of the objects in a set of $5$ objects equals $5!$. Since the order of selection is not important, we divide the number of ways by 5! to get $\dfrac{100\times 98\times 96\times 94\times 92}{5!}=67,\!800,\!320$