There are 20 students in a class with 7 boys . (a) In how many different ways can a team of 2 boys and 6 girls formed ? (b) In how many different ways can a team of 6 be formed such that there is a least one boy and at least one girl in the team ?

1 Answer

a. 36,036 ways
b. 278,460 ways

Explanation:

There are 7 boys and 13 girls in the class.

When picking teams, we're looking at combinations (we don't care in what order the players are picked). The general formula is:

#C_(n,k)=((n),(k))=(n!)/((k!)(n-k)!)# with #n="population", k="picks"#

a

We want a team with 2 boys (from a population of 7) and 6 girls (from a population of 13):

#((7),(2))((13),(6))=21xx1716=36036# ways

b

We want a team of 6 with at least 1 boy and 1 girl. So let's first have 1 guaranteed boy and 1 guaranteed girl:

#((7),(1))((13),(1))#

Now we need to fill in the remaining team. We can pick any of the remaining 18 students to fill in the remaining 4 spots:

#((7),(1))((13),(1))((18),(4))=7xx13xx3060=278460# ways