A problem in my textbook asks:

1. In how many ways can four ours and four girls be arranged in a circle? In how many of these will boys and girls occur alternately?

2. In how many ways can n boys and n girls be arranged in a circle? In how many of these will the boys and girls occur alternately.

The first part of both questions is easy, 7! and (2n-1)!

But I'm really confused on the second parts.

I worked it out as the first boy to sit begins the circle and everyone is placed relative to them. Then the next boy sits can sit to the left, or the right, or opposite the first boy. Then the third has two choices, and the fourth one. And then the girls sit down, the first has a choice of four places to sit, the second three, etc

So I get the answer: (4-1)! * 4! = 144

and for n boys and n girls: (n-1)! n!

but my book says the answer should be 72

and it says for n girls and n boys, the answer is 0.5 n!(n - 1)!

but this doesn't make sense to me, especially as if you only have one boy and one girl then their answer would be half an arrangement, which is nonsense?