Question

A school committee consists of 2 teachers and 4 students. The number of different committees that can be formed from 5 teachers

Answer 1

Option C

2100 different committees that can be formed from 5 teachers  and 10 students

Solution:

A school committee consists of 2 teachers and 4 students

The number of different committees that can be formed from 5 teachers  and 10 students is to be found

This is a combination problem

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 to calculate combinations is:

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

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

Here we have to choose 2 teacher from 5 teacher and 4 student from 10 student

Hence we get,

Number of committees = $5C_2 \times 10C_4$

$\begin{aligned}&5 C_{2} \times 10 C_{4}=\frac{5 !}{2 !(5-2) !} \times \frac{10 !}{4 !(10-4) !}\\\\&5 C_{2} \times 10 C_{4}=\frac{5 !}{2 ! \times 3 !} \times \frac{10 !}{4 ! \times 6 !}\end{aligned}$

To get the factorial of a number n the given formula is used,

$n !=n \times(n-1) \times(n-2) \ldots \times 2 \times 1$

Therefore,

$5 C_{2} \times 10 C_{4}=\frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 3 \times 2 \times 1} \times \frac{10 \times 9 \times \ldots . \times 2 \times 1}{4 \times 3 \times 2 \times 1 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}$

$5 C_{2} \times 10 C_{4}=2100$

Thus 2100 different committees that can be formed from 5 teachers  and 10 students