How many different teams can be selected?

Question

Assume that there are 9 girls and 8 boys in the neighborhood club, and a team of 5 is to be selected.
(1) How many different teams can be selected?

(2) How many different teams can be selected if each team must contain exactly 2 girls and 3 boys?

(3) How many different teams can be selected if each team must contain both boys and girls?

Answer 1

(1)

 

There are a total of 17

 

and  5 have to be selected

the order of selection doesn't matter  as the same 5 will be in a team irrespective of the order in which they are chosen so it's  example of a combination

 

we have to choose 5 from 17   so ¹⁷C₅ =6188 ways

(2) Number of ways of choosing 2 girls from 9= ⁹C₂ = 36

number of ways of choosing 3 boys from 8= ⁸C₃ = 56

so number of ways of choosing 2 girls AND 3 boys = 36 x 56 = 2016



(3) a team containing both boys and girls must have at least 1 boy and 1 girl ..team must have at least one of each gender so that's what I'll do:

Number of ways of choosing 1 boy and 4 girls = ⁸C₁ x ⁹C₄ = 1008

OR Number of ways of choosing 2 boys and 3 girls = ⁸C₂ x ⁹C₃ = 2352

OR number of ways of choosing 3 boys and 2 girls = ⁸C₃ x ⁹C₂ = 2016

OR number of ways of choosing 4 boys and 1 girls = ⁸C₄ x ⁹C₁= 630




OR means ADD



so number of different teams with both boys and girls =1008+2352+2016+630 = 6006 teams