$(10$pts$.)$ Suppose there are $n$ different types of coupons and that each time one collects a coupon, it

is independently of previous coupons collected, a type i coupon with probability $p_i,{\displaystyle \underset{i=1}{\overset{n}{}}}{p}_i=1.$

$($a$)$ Find the expected number of coupons one needs to collect to obtain a complete set of at

least one of each type.

$($b$)$ Repeat $($a$)$ as $p_i=\frac{1}{n}$ for each $i$