Given: $100$ banknotes of $20\$$, $100$ banknotes of $50\$$ and $100$ banknotes of $100\$$. How many ways are to create a pile with the amount of $n$? Notice that you cannot differ between banknotes with the same amount.
I was asked to create a generating function for this problem. The answer is:
$${{{{({x^{20}})}^{101}} - 1} \over {{x^{20}} - 1}} \cdot {{{{({x^{50}})}^{101}} - 1} \over {{x^{50}} - 1}} \cdot {{{{({x^{100}})}^{101}} - 1} \over {{x^{100}} - 1}}$$
I'm not quite sure how to get it. What's the explanation behind this expression?