Let $x_1,x_2,x_3$, and $x_4$ be the numbers glazed, chocolate, sugar, and plain doughnuts that you choose. Then each of these numbers must be a non-negative integer, and
Each solution of $(1)$ in non-negative integers gives you a possible choice of doughnuts, and each possible choice of doughnuts gives you a solution to $(1)$ in non-negative integers. Thus, your problem reduces to counting the solutions to $(1)$ in non-negative integers. This kind of problem is often called a stars-and-bars problem; the linked article gives you the answer,
and a pretty decent explanation of the reasoning behind it. Between what I’ve done here with (a) and what you find in the article, you should be able to make a good stab at (b), but feel free to ping me if you get stuck.