Question

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
A. 10
B. 15
C. 2100
D. 8

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)=(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 * 10C_4`

`\begin{aligned}&5 C_(2) * 10 C_(4)=(5 !)/(2 !(5-2) !) * (10 !)/(4 !(10-4) !)\\\\&5 C_(2) * 10 C_(4)=(5 !)/(2 ! * 3 !) * (10 !)/(4 ! * 6 !)\end{aligned}`

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

`n !=n *(n-1) *(n-2) \ldots * 2 * 1`

Therefore,

`5 C_(2) * 10 C_(4)=(5 * 4 * 3 * 2 * 1)/(2 * 1 * 3 * 2 * 1) * (10 * 9 * \ldots . * 2 * 1)/(4 * 3 * 2 * 1 * 6 * 5 * 4 * 3 * 2 * 1)`

`5 C_(2) * 10 C_(4)=2100`

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