I believe the problem is called counting restricted partitions. (Warning: combinatorialists use the word "partition" in two different ways, and sometimes it's not completely clear from context which one is meant. One must distinguish between partitions of a number, which is the sense meant here, and partitions of a set.) With replacement, the answer is that the number $s_n$ of ways has generating function
$\sum_n s_n x^n = (1 + x^a + x^{2a} + ...)(1 + x^b + x^{2b} + ...)(1 + x^c + x^{2c} + ...)...$
or equivalently
$\frac{1}{(1 - x^a)(1 - x^b)(1 - x^c)...}.$
This is because choosing a term from each factor corresponds to choosing how many times you use each number you can use. For example, with the choices $\{ 1, 2, 5, 10, 25 \}$, this is the problem of counting the number of possible ways to make change for a certain amount of money.
The generating function can be converted into an explicit formula when the set of possibilities is finite, but it is messy in the general case and not really worth caring about. It leads to a straightforward asymptotic which I could explain if you are interested.