Given there are 6 boys and 2 girls arranged in a row, what is the probability there are at least 3 boys separating the 2 girls?

1 Answer
Oct 15, 2017

144 ways

Explanation:

If the 2 girls to be separated by at least 3 boys, that means they can either be separated with 3 boys or with 4 boys. Let's look at each case individually.

Case 1: Separated with 3 boys:

If we have, say, 6 seats numbered 1 through 6, the girls can either be sitting on seats 1 and 5 or on seats 2 and 6. These two cases are just reflections of another, so the total permutations for both will be the same. I'll calculate it once and then multiply by 2 to account for the two ways.

Imagine having girl A and girl B. They can sit in the order of AB or BA. Thus, 2 ways to order the girls.

Now for the boys. There are 4!=24 ways to order them. Thus:

2*2*24=96 ways for Case 1.

Case 2: Separated with 4 boys.

It's the same idea, but there's only one way to seat the girls: seat 1 and seat 6. Because of this, we don't have a 2 in the product.

There are still 2 ways to order the girls and 4!=24 ways for the boys.

2*24=48 ways for Case 2.

Adding up the total number of ways, we get 96+48=144 total ways, assuming each boy and each girl are unique.