# A committee of 4 is to be selected from amongst 5 boys and 6 girls. In how many ways can this be done so as to include

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A committee of 4 is to be selected from amongst 5 boys and 6 girls. In how many ways can this be done so as to include (i) exactly one girl, (ii) at least one girl?

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(i) In this case we have to select one girl out of 6 and 3 boys out of 5.

The number of ways of selecting 3 boys = 5C3 = 5C2 = 10.

The number of ways of selecting one girl = 6C1 = 6.

∴ The required committee can be formed in 6 × 10 = 60 ways.

(ii) The committee can be formed with (a) one boy and three girls, or (b) 2 boys and 2 girls,

or (c) 3 boys and one girl, or (d) 4 girls alone.

The committee can be formed in (a) 5C1 × 6C3 ways ;

The committee can be formed in (b) 5C2 × 6C2 ways ;

The committee can be formed in (c) 5C3 × 6C1 ways ;

The committee can be formed in (d) 6C4 ways.

Hence the required number of ways of forming the committee

= 5C1 × 6C3 + 5C2 × 6C2 + 5C3 × 6C1 + 6C4 = 100 + 150 + 60 + 15 = 325 ways.

(ii) Method II. Required ways = (Committees of 4 out of 11 without any restriction) – (Committees in which no girl is included) = 11C45C4 = 325.