Question

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

Answer 1

If there are `G` girls and `B` boys
and all the boys stand together there are
`color(white)("XXX")(G+1)G!B!` possible arrangements.

Explanation:

If the boys are inserted together in a line of `G` girls,
there are `color(red)("("G+1")")` different places the boys could be inserted each giving a different arrangement.
(One way to see this is to consider how many girls would be to the left of the group of boys; the choices are `{0, 1, 2, ...,G}` for `(G+1)` different possibilities.

The girls could be arranged in `color(blue)(G!)` different sequences:

`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
`color(white)("XXX")color(green)(B!)` different permutations.

So
for each of the `color(red)("("G+1")")` locations where the boys could be inserted in the line of girls
`color(white)("XXX")`there are `color(blue)(G!)` permutations of girsl,
`color(white)("XXX")`and for each of these joint combinations
`color(white)("XXXXXX")`there are `color(green)(B!)` permutations of boys.

Giving `color(red)("("G+1")")*color(blue)(G!)*color(green)(B!)` different permuations.