Question

The principal at Cobb County High School wants to start a mentoring group. He needs to narrow his choice of students to be mentored to six from a group of nine. How many ways can a group of six be selected.

Short answer

The number of ways in which a group of six can be selected is84.

Step-by-step solution

Step 1. Given Information.

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.

Step 2. Calculation.

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:

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

Step 3. Conclusion.

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