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.