Permutation question (1 Viewer)

[asharnadeem]

How many ways can 4 boys and 10 girls be arranged if the boys must not be seated next to any other boys?

 

[quickoats]

Are they sitting in a circle or in a line?

For the case that they're in a line, think of it like this. Make the 10 girls stand in a certain order, and let the boys squeeze in anywhere between 2 girls.

_G_G_G_G_G_G_G_G_G_G_ (10 girls standing in a line - there are 11 spots _ which the boys can take). Think of it like 4 out of the 11 spaces must pick a boy to stand in its place.

There are 10! ways the girls can stand in this formation, and 11P4 positions for the boys. Therefore the total arrangements are 10! x 11P4

The circle has similar reasoning (just harder to illustrate on the computer) but using the spacer method is probably the easiest.

 

[satvik_1008]

you just use the insertion method, so you arrange the girls first (10!) then you insert 4 boys into 11 spaces between the girls (4 choose 11) then arrange the four boys
so your final answer = 10! * 4C11 * 4! = 28740096000

 

[asharnadeem]

Thanks

 

[tickboom]

How many ways can 4 boys and 10 girls be arranged if the boys must not be seated next to any other boys?

You may find this video helpful, where I work through a detailed explanation of one possible approach:

Enjoy!