(i) Boys and girls are alternate.
There can be two posiibilities.
(a)First boy then girl then boy then girl and so on.
It can be done in `5!*5!` ways.
(b) First girl then boy then girl then boy and so on.
It can be done in `5!*5!` ways.
So, for this, there can be `2*5!*5!` ways.
Also, we can write, `(5!)^2 +(5!)^2`
(ii)No two girls sit together.
For this, five girls should be between 5 boys. So, there are 6 places where these 5 girls can be placed.
So, there are `C(6,5)` ways for selecting 5 girls on 6 places.
Now, these `5` girls and `5` boys can be arranged in `5!*5!` ways.
So, there can be `C(6,5)5!5!` ways.
`C(6,5)5!5! = 6*5!5! = 6!5!`
(iii)All the girls sit together.
When all girls sit together there can be `6!` arrangemnts and these 5 girls can be arranged in `5!` ways.
`:.` Total number of arrangements in this case ` = 6!5!`
(iv)All the girls are never together.
Total number of arrangements possible ` = 10!`
Total number of arrangements when all the girls are together `= 6!5!`
`:.` Total number of arrangements when girls are not together `=10
!-6!5!`
