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

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About mvtrinh

Retired high school math teacher.
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