Question

There are 5 teachers and 20 students. Out of them a committee of 2 teachers and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees a particular teacher is included?

Answer 1

The number of teachers = 5

Number of students = 20

The number of ways of selecting 2 teachers from 5 teachers is

= 5C₂ ways.

= `(5!)/(2! xx (5 - 2)!)`

= `(5!)/(2! xx 3!)`

= `(5 xx 4 xx 3!)/(2! xx 3!)`

= `(5 xx 4)/(2 xx 1)`

= 10 ways

The number of ways of selecting 3 students from 20 students is

= 20C₃ 

= `(20!)/(3! xx (20 - 3)!)`

= `(20!)/(3! xx 17!)`

= `(20 xx 19 xx 18 xx 17!)/(3! xx 17!)`

= `(20 xx 19 xx 18)/(3 xx 2 xx 1)`

= 20 × 19 × 3

= 1140 ways

∴ The total number of selection of the committees with 2 teachers and 5 students is

= 10 × 1140

= 11400

A particular teacher is included.

Given a particular teacher is selected.

Therefore, the remaining 1 teacher is selected from the remaining 4 teachers.

Therefore, the number of ways of selecting 1 teacher from the remaining 4 teacher

= 4C₁ ways

= 4 ways

The number of ways of selecting 3 students from 20 students = 20C₃ 

= `(20!)/(3!(20 - 3)!)`

= `(20!)/(3! xx 17!)`

= `(20 xx 19 xx 18  xx 17!)/(3! xx 17!)`

= `(20 xx 19 xx 18)/(3!)`

= `(20 xx 19 xx 18)/(3 xx 2 xx 1)`

Hence the required number of committees

= 1 × 4 × 1140

= 4560