Question

You are getting a line-up ready for a school kickball game. You have

Answer 1

There are 28,800 ways.

There are 10 possibilities for the first child, since it could be a boy first or a girl first.  After this, there are 5 possibilities for the second child (the other gender).  There are 4 choices for the 3rd child, 4 for the 4th child, 3 for the 5th child, 3 for the 6th child, 2 for the 7th child, 2 for the 8th child, 1 for the 9th child, and 1 for the 10th child:
10(5)(4)(4)(3)(3)(2)(2)(1)(1) = 28,800