(a)[3]
Find the number of different ways to arrange the $7$ men in a line so that Ali and Ben are not next to each other.
(b)[3]
Find the number of different ways to arrange the $7$ men and $4$ women in a line with all the men together and all the women together.
(c)[2]
In how many different ways can the $7$ men and $4$ women be split into a group of $6$, a group of $3$ and a group of $2$ with no restrictions?
(d)[4]
The $7$ men and $4$ women are split at random into a group of $6$, a group of $3$ and a group of $2$. Find the probability that Ali, Ben and Charlie are all placed in the same group.