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 most 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 most 3 girls.
The possible cases are the following:
i. Selecting 3 girls and 4 boys
ii. Selecting 2 girls and 5 boys
iii. Selecting 1 girl and 6 boys
iv. Selecting 7 boys
Let us assume the no. of ways of selecting is N2.
⇒ N2 = ((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 2 girls out of 4 girls) × (no. of ways of selecting 5 boys out of 9 boys)) × ((no. of ways of selecting 1 girls out of 4 girls) × (no. of ways of selecting 6 boys out of 9 boys))
⇒ N2 = ((4C3) × (9C4)) + ((4C2) × (9C5)) + ((4C1) × (9C6)) + (9C7)
We know that ,
And also n! = (n)(n – 1)......2.1
⇒
⇒
⇒
⇒ N2 = (4 × 126) + (6 × 126) + (4 × 84) + (36)
⇒ N2 = 504 + 756 + 336 + 36
⇒ N2 = 1632
The no. of ways of selecting 7 members with at most 3 girls is 1632.