There is a group of 20 people: 15 girls and 5 boys. We want to pick one person f

Question

There is a group of 20 people: 15 girls and 5 boys. We want to pick one person for our cooking group, one person for our paintball group, one person for our tennis group, and one for our art group.

a) In how many different ways can this be done?

b) In how many different ways can we do this if we want to have at least one boy?

c) In how many different ways can we do this if we want to have exactly one boy?

d) In how many different ways can we do this if we want to have at least one boy and one girl?

Explanation / Answer

Total number of people= 20; Number of girls=15; number of boys=5

a) Imagine this problem as 2 separate halves:

So, number of ways this can be done is = 20C4 * 4! = 116280

b) Similarly, different ways of doing this with our selection having at least1 boy (this changes the first step in part A)

= Total number of ways of selecting 4 people - No. of ways of doing this with no boy present in our selection

= (20C4 - 15C4) * 4! = 83520

c) Similarly, different ways of doing this with our selection having exactly 1 boy (this changes the first step in part A)

= (15C3*5C1) * 4! = 54600

d) Number of ways of selecting 4 people with at least one boy and one girl = (15C1*5C1*18C2) *4!=275400

[The logic behind calculating Part D is that we select 1 boy from 5 boys and 1 girl from 15 girls thus satisfying the fact that there is at least 1 of each gender in your selection. Now there are 2 spots left to fill and 18 students to choose from, so randomly choose 2 students from 18 and arrange all 4 in 4! ways]

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