# Generating function to word problem.

So I've done several of these from sequences to functions and vice versa but I can seem to get this one.

Find a generating function for the given series.

The sequence $$a_0 , a_1 , ....$$ where $$a_n$$ is the number of ways to give a player n dollars using only 5 dollar red poker chips, 10 dollar blue poker chips, 25 dollar green poker chips, and 100 dollar black poker chips.

I tried using diffrent variables n it was a mess I also tried putting the numbers into the expansion but it didn't work out right.

## 1 Answer

HINT This is a classic that is discussed by Wilf.

If we need a generating function to give $n$ dollars using just \$5 chips, it is $$1 + x^5 + x^{10} + \ldots = \frac{1}{1-x^5}$$ and using \$10 it would be $$1 + x^{10} + x^{20} + \ldots = \frac{1}{1-x^{10}}$$ and using both \$5 and \$10 chips it is $$\left( \sum_{k=0}^\infty x^{5k} \right) \left( \sum_{k=0}^\infty x^{10k} \right) = \frac{1}{\left(1-x^5\right)\left(1-x^{10}\right)}$$ Can you finish the problem?

• Yeah I was putting the 5,10 etc in the wrong spots would be a giant pain to solve that with all those terms with partial fractions. – Faust Mar 20 '17 at 20:49
• @Faust7: We weren't asked to solve it with partial fractions. An expression like the right side of the last equation is a fine answer. – Ross Millikan Mar 20 '17 at 21:36
• That's why I said would ^^ can't belive I was trying to put the 5s with the x term.... – Faust Mar 20 '17 at 21:40