Question 1095609
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. 
<pre>
BBBBBBBB
 GGGGGG

Answer = 8! x 6!
</pre>
b) in a single row, if the girls must be kept together. 
<pre>
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!
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c) in a single row, if no two girls should stand next to each other.
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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</pre>