Question

How many different committees can be formedfrom 6 teachersand 42 students if the committee consists of 3 teachers and 3 students?The committee of 6 members can be selected in different ways.

Answer 1

Answer

229,600ways

Given

The committee of six members consists of 3 teachers and 3 students

Solution

We need to choose 3 of the 6 teachers AND 3 of the 42 students:

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

where the combination is a selection of r possible combinations of objects from a set of n objects.

$\begin{gathered} (^6C_3)(^{42}C_3)=\frac{6!}{(6-3)!\times3!}\times\frac{42!}{(42-3)!\times3!} \\ \\ \frac{6!}{(6-3)!\times3!}\times\frac{42!}{(42-3)!\times3!}=\frac{6!^{}^{}^{}}{3!\times3!}\times\frac{42!}{39!\times3!} \\ \\ \frac{6!^{}^{}^{}}{3!\times3!}\times\frac{42!}{39!\times3!}=20\times11480 \\ \\ \\ =229600 \end{gathered}$

The Final Answer

The committee of 6 members can be selected in 229,600 different ways.