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