determine the number of 5 student committees that can be formed if?

Question

a) the committee must have 4 boys?

 

b) the committee must have 1 or 2 girls?

 

c) there must be at least one boy on the committee?

Answer 1

To find the number of ways you can form a subset of your group with subgroups (boys and girls in this case) you can find the combinations of each subgroup, multiply them, and divide by the combinations of the whole group.

 

22 in the overall group

12 girls

10 boys

 

a) a) the committee must have 4 boys?

You want 4 boys, 1 girls:

 

4 out of 10 boys, 1/12 girls, 5/22 total: ₁₀C₄ * ₁₂C₁ / ₂₂C₅

 

 

b) the committee must have 1 or 2 girls?

The answer to a) is for 1 girl

For two girls (and 3 boys), you can do the same sort of thing, then add the two together.

3/10 boys, 2/12 girls, 5/22 total:  ₁₀C₃ * ₁₂C₂ / ₂₂C₅

 

c) there must be at least one boy on the committee?

This means there is 1 or 2 or 3 or 4 or 5 boys. You could calculate all 5 probabilities and add them up, but it's a lot easier to calculate the complement (opposite). The complement of at least one boy is no boys.

 

0/10 boys, 5/12 girls, 5/22 total: can you find this value?

 

A probability and its complement must add up to 1, so to find at least one boy, subtract no boys from 1.

1 - P(no boys) = P(at least one boy)

 

I hope this helps. If you have more questions, please comment!