Question

How many ways a committee of six persons from 10 persons can be chosen along with a chair person and a secretary?

Answer 1

Number of persons = 10

Number of committee members to be selected = 6

A committee must have a chairperson and a secretary, the remaining 4 members.

The number of ways of selecting a chairperson and a secretary from 10 persons is

= ¹⁰C₂

= `(10!)/(2!(10 - 2)!)`

= `(10!)/(2!  8!)`

= `(10 xx 9 xx 8!)/(2! xx 8!)`

= `(10 xx 9)/(2 xx 1)`

= 5 × 9

= 45

After the selection of chairperson and secretary remaining number of persons = 8

Number of ways of selecting remaining 4 committee members from the remaining 8 persons

= ⁸C₄ 

= `(8!)/(4!(8 - 4)!)`

= `(8!)/(4! xx 4!)`

= `(8 xx 7 xx 6 xx 5 xx 4!)/(4! xx 4!)`

= `(8 xx 7 xx 6 xx 5)/(4!)`

= `(8 xx 7 xx 6 xx 5)/(4 xx 3 xx 2 xx 1)`

= 2 × 7 × 5

= 70

∴ A number of ways of selecting the six committee members from 10 persons.

= 45 × 70

= 3150 ways