Given that a group consists of 4 girls and 7 boys.

We need to select a team of 5 members with the following conditions:

i. If a team has no girl

ii. If a team has at least one boy and one girl

iii. If a team has at least 3 girls.

i. Given that we need to select a team of 5 members with no girl present in it out of 4 girls and 7 boys.

Let us assume the no. of ways of selection be N

⇒ N = (selecting 5 members out of 7 boys without any girl)

⇒ N = ⁷C₅

We know that ,

And also n! = (n)(n – 1)......2.1

⇒

⇒

⇒

⇒ N = 21 ways

The no. of ways of selecting 5 members without a girl is 21 ways.

ii. Given that we need to select team of 5 members with at least 1 boy and 1 girl.

Let us assume the no. of ways of selection be N₁.

⇒ N₁ = (Total ways of selecting 5 members out of all 11 members) – (No. of ways of selecting 5 members without any girl)

⇒ N₁ = (¹¹C₅)–(⁷C₅)

We know that ,

And also n! = (n)(n – 1)......2.1

⇒

⇒

⇒

⇒ N₁ = 462–21

⇒ N₁ = 441

The no. of ways of selecting 5 members with at least 1 girl and 1 boy is 441.

iii. Given that we need to find the no. of ways to select 5 members with at least 3 girls out of 7 boys and 4 girls.

The following are the possible cases:

i. Selecting 3 girls and 2 boys

ii. Selecting 4 girls and 1 boy

Let us assume the no. of ways of selection be N₂.

⇒ N₂ = (No. of ways of selecting 3 girls and 2 boys out of 7 boys and 4 girls) + (No. of ways of selecting 4 girls and 1 boy out of 7 boys and 4 girls)

⇒ N₂ = ((⁴C₃) × (⁷C₂)) + ((⁴C₄) × (⁷C₁))

We know that ,

And also n! = (n)(n – 1)......2.1

⇒

⇒

⇒

⇒ N₂ = (4 × 21) + (1 × 7)

⇒ N₂ = 84 + 7

⇒ N₂ = 91

The no. of ways of selecting 5 members with at least 3 girls is 91.