How many different ways can five boys and five girls form a circle
with boys and girls alternate?
First we find the number of ways to place the boys in a circle.
Suppose the boys are A,B,C,D,E. Then since a circle can be turned at
any angle without changing the circle, we can think of, say A, being
fixed, say facing north, and we simply arrange the 4 boys B,C,D,E
around him.
So we can place the 4 boys in these 4 positions,
1. the first position clockwise from A,
2. the first position counter-clockwise from A,
3. the second position clockwise from A,
4. the second position counter-clockwise from A,
So we can place the four boys around A in 4! ways.
[We do not include cases of putting 4 boys arranged around B,
or around C, etc., because they are all counted among the 4!
because they are just rotations of the same ones where the
four were arranged around A.]
Now for each of the 4! ways to arrange the boys, there are 5
positions to put the 5 girls in. That's 5! ways. Therefore
Answer: 4!5! = 24(120) = 2880 ways.
Edwin
