Chapter 9.6, Problem 60E

To determine

To find: How many different committees are possible from the 100 U.S senators.

Answer to Problem 60E

  $4.42\times {10}^{16}$ different committees are possible

Explanation of Solution

Given information:

Consider that the U.S Senate Committee of 14 members is to be formed from 100 U.S senators.

In order to find a 14 membersCommittee from 100 U.S senators, when order does not matter, we use combinations.

The number of combinations is given by:

  $\left(\begin{array}{c}n\\ k\end{array}\right)=\frac{n!}{k!\left(n-k\right)!}$

Where $n$ is the number of U.S senators and $k$ is the number of members to be picked.

Therefore,

  $\begin{array}{l}n=100\\ k=14\end{array}$

On substituting values in above combinations formula

We get

  $\begin{array}{l}\left(\begin{array}{c}n\\ k\end{array}\right)=\frac{n!}{k!\left(n-k\right)!}\\ \Rightarrow \left(\begin{array}{c}100\\ 14\end{array}\right)=\frac{100!}{14!\left(100-14\right)!}\\ \Rightarrow \left(\begin{array}{c}100\\ 14\end{array}\right)=\frac{100!}{14!\left(86\right)!}\\ \Rightarrow \left(\begin{array}{c}100\\ 14\end{array}\right)=\frac{100\times 99\times 98\times \mathrm{...}\times 87\times 86!}{14!\left(86\right)!}\\ \Rightarrow \left(\begin{array}{c}100\\ 14\end{array}\right)=\frac{100\times 99\times 98\times \mathrm{...}\times 87}{14\times 13\times 12\times 11\times \mathrm{...}\times 2\times 1}\\ \Rightarrow \left(\begin{array}{c}100\\ 14\end{array}\right)=4.42\times {10}^{16}\end{array}$

Therefore, $4.42\times {10}^{16}$ different committeesare possible.