91Ó°ÊÓ

Given, the principal at Cobb County High School wants to start a mentoring group and he needs to narrow his choice of students to be mentored to six from a group of nine. The number of ways in which a group of six can be selected 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 arrangement and the number of combinations is given by $C\left(n,r\right)=\frac{n!}{\left(n-r\right)!r!}$

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

Here n=9,r=6

Plugging the values:

$\begin{array}{l}C\left(9,6\right)=\frac{9!}{\left(9−6\right)!6!}\\ C\left(9,6\right)=\frac{9!}{3!6!}\\ C\left(9,6\right)=\frac{9â‹\dots 8â‹\dots 7â‹\dots 6!}{3â‹\dots 2â‹\dots 1â‹\dots 6!}\\ C\left(9,6\right)=\frac{504}{6}\\ C\left(9,6\right)=84\end{array}$

Hence, the number of ways in which a group of six can be selected is 84.