Please carefully explain your answer for each problem. Thank you.

Find a regular expression corresponding to each of the following subsets of $\{a,b{\}}^{*}.$

$($a$)$ The language of all strings containing exactly two a$\text{'}$s$.$

$($b$)$ The language of all strings containing at least two a$\text{'}$s$.$

$($c$)$ The language of all strings that do not end with ab$.$

$($d$)$ The language of all strings that begin or end with aa or bb$.$

$($e$)$ The language of all strings not containing the substrings aa$.$

$($f$)$ The language of all strings in which the number of a$\text{'}$s is even.