A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of :
at least 3 girls?
Given that we need to select 7 members out of 9 boys and 4 girls by following the conditions:
i. exactly 3 girls
ii. at least 3 girls
iii. at most 3 girls.
It is told we need to select 7 members out of 9 boys and 4 girls with at least 3 girls.
The possible cases are the following:
i. Selecting 3 girls and 4 boys
ii. Selecting 4 girls and 3 boys
Let us assume the no. of ways of selecting is N1.
⇒ N1 = ((no. of ways of selecting 3 girls out of 4 girls) × (no. of ways of selecting 4 boys out of 9 boys)) × ((no. of ways of selecting 4 girls out of 4 girls) × (no. of ways of selecting 3 boys out of 9 boys))
⇒ N1 = ((4C3) × (9C4)) + ((4C4) × (9C3))
We know that ,
And also n! = (n)(n – 1)......2.1
⇒
⇒
⇒
⇒ N1 = (4 × 126) + (1 × 84)
⇒ N1 = 504 + 84
⇒ N1 = 588
The no. of ways of selecting 7 members with at least 3 girls is 588.