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

1 Answer
Jan 31, 2016

Answer:

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.