How many ways can five members of the 100-member United States Senate be chosen to be put on a committee?

Given to choose five members of the 100-member United States Senate to be put on a committee. The number of ways the committee can be chosen is to be determined.

A permutation is when n objects are available and r are to be picked and arranged in a certain order and the number of permutations is given by $P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$.

A combination is when n objects are available and r are to be picked without arrangementand the number of combinations is given by $C\left(n,r\right)=\frac{n!}{\left(n-r\right)!r!}$.

Here, the order of choosing doesn’t affect the final outcome. Hence the employees are just picked and not arranged i.e. the given situation is a combination.

Here n=100,r=5

Plugging the values:

$\begin{array}{l}C\left(100,5\right)=\frac{100!}{\left(100-5\right)!5!}\\ C\left(100,5\right)=\frac{100\cdot 99\cdot 98\cdot 97\cdot 96\cdot 95!}{95!\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}\\ C\left(100,5\right)=\frac{100\cdot 99\cdot 98\cdot 97\cdot 96}{5\cdot 4\cdot 3\cdot 2\cdot 1}\\ C\left(100,5\right)=\frac{9034502400}{120}\\ C\left(100,5\right)=75287520\end{array}$

Hence, the number of ways the committee can be chosen is 75287520.