A debating team consisting of 5 persons is to be chosen from a group of 7 boys and 5 girls. In how many ways can this team be formed so that it contains,

(2) at least one girl

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A debating team of 5 persons has to be created from a group of 7 boys and 5 girls.

The number of ways of creating a team with at least one girl is:

C(5, 1)*C(7, 4) + C(5, 2)*C(7, 3) + C(5, 3)*C(7, 2) + C(5, 4)*C(7, 1) + C(5, 5)*C(7, 0)

= 175 + 350 + 210 + 35 + 1

= 771

**There are 771 ways of creating a team with at least one girl.**

We have 7 boys and 5 girls. So the teams we form can be categorized in to following forms.

- Teams with girls and boys
- Teams with only boys
- Teams with only girls

So except boys only team, every other team has a girl.

Teams containing at least one girl

= Total teams-teams with only boys

`= ^12C_5-^7C_5`

`= 792-21`

`= 771`

*So we can have 771 teams at least a girl included.*

**Sources:**

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