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 student is excluded?

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 student is excluded.

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

= ⁵C₂ 

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

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

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

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

= 5 × 2

= 10

A particular student is excluded

∴ The number of remaining students = 19

Number of ways of selecting 3 students from 19 students

= ¹⁹C₃ 

= `(19!)/(3! xx (19 - 3)!)`

= `(19!)/(3! xx 16!)`

= `(19 xx 18 xx 17 xx 16!)/(3! xx 16!)`

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

= `(19 xx 18 xx 17)/(3 xx 2 xx 1)`

= 19 × 3 × 17

= 969

∴ The required number of committees

= 10 × 969

= 9690