number of ways to make $2.00 How many different ways can you make $2.00 using only 1 cent, 5 cent, 10 cent, and 25 cent pieces, and 1 and 2 dollar bills (there are 100 cents in a dollar)? I have worked out an equation:
$$p + 5n + 10d + 25q + 100l + 200t = 200$$
Where all variables are non-negative integers. But I don't know what to do from here to figure out how many different sets of non-negative integers there are that satisfy this equation. What do I do next?

Use generating functions.
All you have to do is find the coefficient of $x^{200}$ in the formula
$$
(1 + x + x^2 + \cdots)(1 + x^5 + x^{10} + \cdots)(1 + x^{10} + x^{20} + \cdots)
$$
$$
\cdot (1 + x^{25} + x^{50} + \cdots)(1 + x^{100} + x^{200} + \cdots)(1 + x^{200} + x^{400} + \cdots)
$$
The problem is inherently quite computation-heavy, but reducing it to the question above makes the computation much more straightforward, and also very easy to do with a computer.
