Question

A committee must be formed with 3 teachers and 7 students. If there are 7 teachers to choose from, and 9 students, how many different ways could the committee be made?

Answer 1

Step 1

The question is combinations because it involved selection.

$^nC_r\text{ = }\frac{n!}{(n\text{ - r)!r!}}$

Step 2

Number of ways of selecting 3 teachers out of 7

$\begin{gathered} n\text{ = 7, r = 3} \\ ^7C_3\text{ = }\frac{7!}{(7\text{ - 3)!3!}} \\ =\text{ }\frac{7\times6\times5\times4!}{4!\times3\times2\times1} \\ =\text{ 35 } \end{gathered}$

Step 3

Number of ways of selecting 7 students out of 9

$\begin{gathered} n=\text{ 9 , r = 7} \\ ^9C_7\text{ = }\frac{9!}{(9\text{ - 7)!7!}} \\ =\frac{9!}{2!7!} \\ =\text{ }\frac{9\times8\times7!}{2\times1\times7!} \\ \text{= 9 }\times4 \\ =\text{ 36} \end{gathered}$

Final answer

Different ways could the committee be made

= 35 x 36

= 1260 ways