Number of combinations to exactly sum to a value of 200

Question

You have coins of 1, 2, 5, 10, 20, 50, 100 and 200 pennies.

One combination could be 1*10 + 5*18 + 10*5 + 1*50 = 200. The order doesn't matter.

Does anyone knows a way to solve this problem by using combinatorial analysis?

Answer 1

It should be the coefficient of $x^{200}$ in the following expression. [Update: Thanks to the tip from @whuber, it can be simplified using partial fractions. I will try to add more steps later today after work.]

$$\frac{1}{(1-x)(1-x^2)(1-x^5)(1-x^{10})(1-x^{20})(1-x^{50})(1-x^{200})}$$