In what ways 2 Men and 3 ladies and 2 childern can be sit so that any of ladies and any of childern do not sit together?(however any childern and any lady can sit together)
I have solved above question with a different method but I am confused how to solve this with gap method or slot method
Please help
All problems can't necessarily be straight-jacketed into a particular mould, here's one way.
The men and children can be placed in $4! = 24$ ways,
of which, by symmetry, $12$ will be with the children together, and $12$ with the children apart.
If the children are together, eg $\bullet MCCM\bullet\;\; or\;\; \bullet M\bullet MCC$,
$2$ gaps will be available for placing the first woman, and gaps will increase with each placement, so the women can be placed in $2\cdot3\cdot4 = 24$ ways.
And if the children are apart, eg$\;CM\bullet MC$ or $\bullet MCMC$,
only $1$ gap will be available for the first woman, so $1\cdot2\cdot3 = 6$ ways
Putting the pieces together, answer = $12(24+6) = 360$ ways