# How would you use permutations to find the arrangements possible if a line has all the boys stand together?

##### 1 Answer

If there are

and all the boys stand together there are

#### Explanation:

If the boys are inserted together in a line of

there are

(One way to see this is to consider how many girls would be to the left of the group of boys; the choices are

The girls could be arranged in

#G# choices for the first position;

#color(white)("XX")(G-1)# for the second (once the first has been determined)

#color(white)("XXXX")(G-2)# for the third (once the first two have been determined)

#color(white)("XXXXXX")(G-3)# for the fourth...

and so on, until ...

#color(white)("XXXXXXXXXXXX")2# for the second last position

#color(white)("XXXXXXXXXXXXX")1# for the last position.

For a combination of

#color(white)("XXX")G xx (G-1) xx (G-2) xx (G-3) xx ... xx 2 xx1 = G!# different permutations.

Similarly the boys could be arranged in

So

for each of the

Giving