91Ó°ÊÓ

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:

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

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