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?
 A: Recursive algorithm for general monetary systems:
Let $f(x;a_0,a_1...a_k)$ be the number of ways to make $x$c with coins/notes of value $a_i$.  
$f(200;1,5,10,25,100,200)=1+f(200;1,5,10,25,100)$
$f(200;1,5,10,25,100)=1+f(100;1,5,10,25)+f(200;1,5,10,25)$
$f(200;1,5,10,25)=1+f(25;1,5,10)+...+f(200;1,5,10)$ 
Carry on like this, using $f(0;...)=1$ and 
$f(x;a_0..a_k)=1+f(a_k;...a_{k-1})+...+ f(na_k;...a_{k-1})$
where $n$ is the greatest integer such that $na_k \leq x$.
A: 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.
A: The answer is $~2729~$ ways. 
Answered with the help of a computer program made by me.
A: This is essentially the coin changing problem popularized by Pólya in his "On Picture-Writing", AMM 63(10), 689-697 (dec 1956). Graham, Knuth and Patashnik's "Concrete Mathematics" develops a closed form solution.
