A school teacher has 6 girls and 8 boys to arrange for a choir. Determine the number of ways she can arrange the 14 children,
a) if the boys should stand behind the girls.
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Answer = 8! x 6!
b) in a single row, if the girls must be kept together.
We can choose either 0,1,2,3,4,5,6,7, or all 8 boys to place
to the left of the row of 6 girls.
GGGGGGBBBBBBBB
BGGGGGGBBBBBBB
BBGGGGGGBBBBBB
BBBGGGGGGBBBBB
BBBBGGGGGGBBBB
BBBBBGGGGGGBBB
BBBBBBGGGGGGBB
BBBBBBBGGGGGGB
BBBBBBBBGGGGGG
That's 9 ways to arrange them gender-wise, as you see above.
Answer = 9 x 6! x 8!
c) in a single row, if no two girls should stand next to each other.
We line up the boys with one space on both sides of each boy.
_B_B_B_B_B_B_B_B_
There are 9 spaces between, before or after the boys available to
place the 6 girls in. That's
9 choose 6 = 9C6 ways to arrange them gender-wise.
Answer = 9C6 x 8! x 6!
Edwin