We will use generating functions to determine how many ways there are to use pennies, nickels, dimes, quarters, and half-dollars to give n cents change.

(a) Write the generating function P(x) for the number of ways to use only pennies to make n cents.

(b) Write the generating function N(x) for the number of ways to use only nickels to make n cents.

(c) Write the generating function for the number of ways to use only nickels and pennies to change n cents.

(d) Write the generating function for the number of ways to use pennies, nickels, dimes, quarters, and half-dollars to give n cents change.

(e) Explain how to use this function to find out how many ways are there to change 50 cents; you do not have to provide the answer or actually carry out the process.