We are interested in generating functions for the number of different ways to compose a bag of n donuts subject to various restrictions. For each of the restrictions below, find a closed form for the corresponding generating function.

(a) All the donuts are chocolate and there are at least 3.

(b) All the donuts are glazed and there are at most 2.

(c) All the donuts are coconut and there are exactly 2 or there are none.

(d) All the donuts are plain and their number is a multiple of 4.

(e) The donuts must be chocolate, glazed, coconut, or plain with the numbers of each flavor subject to the constraints above.

(f) Now find a closed form for the number of ways to select n donuts subject to the above constraints.