Question

A student dance committee is to be formed consisting of 2 boys and 4 girls. If the membership is to be chosen from 5 boys and 6 girls, how many different committees are possible?

Answer 1

150 different committees are possible

Solution:

Given that a student dance committee is to be formed consisting of 2 boys and 4 girls

The membership is to be chosen from 5 boys and 6 girls

To find : number of different possible committees

A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected

The formula for combination is given as:

$n C_{r}=\frac{n !}{(n-r) ! r !}$

where "n" represents the total number of items, and "r" represents the number of items being chosen at a time

We have to select 2 boys from 5 boys

So here n = 5 and r = 2

$\begin{aligned} 5 C_{2} &=\frac{5 !}{(5-2) ! 2 !}=\frac{5 !}{3 ! 2 !} \\\\ 5 C_{2} &=\frac{5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1 \times 2 \times 1} \\\\ 5 C_{2} &=10 \end{aligned}$

We have to select 4 girls from 6 girls

Here n = 6 and r = 4

$\begin{aligned} 6 C_{4} &=\frac{6 !}{(6-4) ! 4 !}=\frac{6 !}{2 ! 4 !} \\\\ 6 C_{4} &=\frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 4 \times 3 \times 2 \times 1}=15 \end{aligned}$

Committee is to be formed consisting of 2 boys and 4 girls:

So we have to multiply $5 C_{2}$ and $6 C_{4}$

$5 C_{2} \times 6 C_{4}=10 \times 15=150$

So 150 different committees are possible