Question

You are getting a line-up ready for a school kickball game. You have 4 girls and 4 boys. The rules state each child must kick the same number of times and alternate girl-boy or boy-girl. How many ways can a line-up be made for one round of kicking?

Answer 1

Answer:

So the total number of  ways  in which the line-up can be made is 2880 ways.

Step-by-step explanation:

Please refer the attached figure for reference.

Now, there are 4 girls and 4 boys and  we are asked to find the number of  ways in which they can be lined up alternately.

That basically means that there should be a boy next to a girl in the arrangement.

So, at first we will arrange the 4 boys ( at B positions in the figure ), and that can be done in 4! ways.

Now, there are 5 positions ( 5 blue dots in the figure ), which can be taken by 4 girls, in such a way that the boys and girls are placed alternately.

So, 4 girls can take 5 positions in $5_P__4$ ways.

So the total number of  ways  in which the line-up can be made = 4! x $5_P__4$ = 24 x 120 = 2880 ways.