Counting with Generating Functions

[tawdry anchor]

Hey all! I'm having trouble wrapping my mind around counting with generating functions. I think I get the basic idea, but once things more past the most basic examples, I start to have trouble.

Here's the question I'm current working on, but an all around better understanding would be most helpful:

How many ways are there to distribute 20 cents to n children and 1 parent if the parent receives either a nickel or a dime and each child can receive any amount?

To start, I'd like to ask: is this at least the correct initial representation?

[tawdry anchor]

Help?

[delicate oxide]

Isn’t The X^10 meant to be a (1-x^10) so that the (1-x^5) can be divided out via difference of squares

[tawdry anchor]

I don't know what you mean by difference of squares

But you're right, I caught that mistake

I've tried this, thus far, but I'm still not seeing any real path forward