
Change combinatorix
So i know I need to use combinatorix for this, but I'm not exactly sure how.
4.3.19 In how many ways can you make change for a dollar, using pennies, nickels, dimes, quarters, and halfdollars? For example, 100 pennies is one way; 20 pennies, 2 nickels, and 7 dimes is another. Order doesn't matter.
Thanks in advance!

Did you try generating functions?
Let be the number of nonnegative solutions to,
.
Create the generating function,
.
Now, realize that,
Where,
Thus, by geometric series,
Does that work out?

There is no easy way to do this problem!
The most efficient way is by generating functions.
Can you expand the following expression:
?
If you can then the coefficient of is the answer to the question.

The highest term I got when multiplying all of that together was (3x^19)/(20(x^20x^15+x^10x^5+1)). Did I do something wrong?

I'm still stuck on this one. I don't get an x^100 term when I do the product of the geometric series. What do I do?