SOLUTION: We are to seat 5 boys, 5 girls, and 1 parent in a circular arrangement. In how many ways can this be done if no boy is to sit next to a boy and no girl is to sit next to a girl? Wh
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Question 344851: We are to seat 5 boys, 5 girls, and 1 parent in a circular arrangement. In how many ways can this be done if no boy is to sit next to a boy and no girl is to sit next to a girl? What if there are 2 parents?
I really just don't know where to start. I've drawn it up, but if you count symmetries and assume the boys and girls to be indistinguishable its weird.. i'm honestly just not good at this haha. thanks for all your help though Answer by scott8148(6628) (Show Source):
the only way for no two boys or two girls to be next to each other is for them to be alternating around the circle
imagine a ring of 5 boys (with spaces) and a ring of 5 girls (with spaces)
___ the rings are intermeshed to make the seating arrangement
how many ways can five things be ordered or arranged sequentially?
___ usually , 5!
___ but a circular arrangement repeats itself as many times as there are positions on the circle
___ so there are 4! (5! / 5) orders for the ring of boys and for the ring of girls
___ there are also 5 ways for the rings to intermesh
so the number of boy-girl arrangements is ___ 4! * 4! * 5
one parent can insert a chair at any of 10 places on the circle
___ so boys and girls and one parent is ___ 4! * 4! * 5 * 10
the second parent would have 11 places to insert a chair
___ so boys and girls and two parents is ___ 4! * 4! * 5 * 10 * 11
the second parent would also allow an adjacent boy and girl to swap positions
so that two boys and two girls
would be separated by parents instead of other boys and girls
___ 10 possible swaps with two positions for the parents
___ the possible arrangements now becomes ___ 4! * 4! * 5 * (10 * 11 + 20)
