Find the number of ways in which 5 girls and 5 boys can be arranged in row
(i) if no two boys are together.
(ii) if boys and girls are alternate.
(iii) all the girls sit together and all the boys sit together.
(iv) all the girls are never together.

Explanation

(i) Here there is no any condition for arranging the girls.
Now 5 girls can be arranged in 5! Ways.
`xx GxxGxxGxxGxxGxx`
When girls are arranged, six gaps are created as shown in figure with `xx`.
Now boys must occupy the places with `xx` marked so that no two boys are together.
Five boys can be arranged in these six places in `.^(6)P_(5)` ways.
Hence total number of arrangements is `5!xx .^(6)P_(5)`
(ii) Let us first arrange five girls among themselves.
Number of ways is 5!.
Since girls and boys are alternate, we have following places marked with `xx` for boys.
`xx G xx GxxGxxGxxG`
or `GxxGxxGxxGxxGxx`
In each of the above cases boys can arrange among themselves in 5! ways.
Hence, total number of arrangements
`=5!xx5!+5!xx5!=2xx5!xx5!`
(iii)
Since all boys are together and all girls are together, we can conisder boys and girls as two units.
These two units can be arranged in 2! ways.
Also, in each unit five persons can be arranged in 5! ways.
So, total number of arrangements are `2!xx5!xx5!=2xx(5!)^(2)`
(iv) Number of arrangements that all girls are not together =total arrangement without any restrictions -arrangement when all girls are together
`=(12)!-7!xx6!`