# In how many ways can 7 boys and 7 girl be lined up if a girl must be first in line and girls and boys alternate positions in line.

jeew-m | Certified Educator

Lets take girls as G and boys as B

G,B,G,B,G.......G,B.

The above is how the arrangement will appear to be.

The number of ways in which the girls can be arranged `= 7!`

For each such arrangement the ways in which the boys can be arranged `=7!`

Therefore the total number of arrangements `=(7!)^2`

aruv | Student

Let G stand for girl  and B stand for boy

Thus we have two cases

(i) countng starts from left

GBGBGBGBGBGBGB

In this arrangment  girls can be arrange in 7! and boys can be arranged  7!

Thus by fundamental principle  total no. of arrangements are

`7! xx 7! = (7!)^2`

`` (ii) counting starts from right

BGBGBGBGBGBGBG

In this arrangment  girls can be arrange in 7! and boys can be arranged  7!

Thus by fundamental principle  total no. of arrangements are

`7! xx7! =(7!)^2`

Thus total number of possible arrangements are

`(7!)^2+(7!)^2=2xx(7!)^2`