How would you use permutations to find the arrangements possible if a line has all the boys stand together?
If there are
and all the boys stand together there are
If the boys are inserted together in a line of
(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
for each of the