A committee is to be chosen from a set of $7$ women and $4$ men.

$($a$)$ How many ways are there to form the committee if the committee has $5$ people, $3$ women and $2$ men?

$($b$)$ How many ways are there to form the committee if the committee can be any positive size but must have equal numbers of women and men?

$($c$)$ What is the probability Mr$.$ Baggins is chosen if the committee consists of exactly $4$ people?

$($d$)$ Suppose the committee has $4$ people, two of each sex. What is the probability that at most one of Mr$.$ and Mrs$.$ Baggins is chosen to be on the committee?.