using 10 and 20 dollar bills?

its a generating function problem, for the life of me i dont know what to do.

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- May 8th 2008, 07:47 AMp00ndawghow many ways to make change for 100 dollars?
using 10 and 20 dollar bills?

its a generating function problem, for the life of me i dont know what to do. - May 8th 2008, 08:06 AMPlato
The answer is the coefficient of $\displaystyle x^{100}$ in the expansion of $\displaystyle \left( {\sum\limits_{k = 0}^{10} {x^{10k} } } \right)\left( {\sum\limits_{k = 0}^5 {x^{20k} } } \right)$.

Can you finish? - May 8th 2008, 08:08 AMMoo
- May 8th 2008, 08:24 AMPlato
- May 8th 2008, 08:25 AMMoo
- May 8th 2008, 08:26 AMMathnasium
Maybe this isn't really the way you're supposed to do it, but can't we just reason that, when making change for $100, we can use either 0, 1, 2, 3, 4, or 5 $20 bills, with the rest being $10 bills, so there are 6 ways?

(Specifically, for ($10, $20): (10, 0), (8, 1), (6, 2), (4, 3), (2, 4), and (0, 5)) - May 8th 2008, 08:49 AMjanvdl
- May 8th 2008, 09:21 AMp00ndawg
thanks for all the help, this question was actually on my final and I ended up doing what you say. Because when I took the test earlier this morning i just couldnt figure out any other way to do it.

but i just wanted to actually know the mathematical way to do it.

Mr. plato gave the way im assuming my teacher wanted it, but I totally did not expect that question, I think the answer I got was right (6), but im not sure if he'll approve of my methods lol.(Rofl)

So i think i may have gotten it right. - May 8th 2008, 09:23 AMjanvdl
- May 8th 2008, 09:25 AMp00ndawg
- May 8th 2008, 09:36 AMPlato
It could be that the instructor expected the exact answer the way I gave it.

I have asked students to set up such problems just to check on understanding.

Unless one spends a good deal of time on how coefficients are calculated from generating functions, I don’t think we can expect much more than that. There is in fact a whole textbook on generating functions, it could be a whole course. - May 8th 2008, 09:48 AMp00ndawg
yea he might have, and I did indeed go through the whole coefficient process of figuring it out. He's is kind of lenient at times, and im hoping he'll give it to me because the test was fairly rough.

There was one other curveball question he put on the test that I didnt even THINK of reviewing, my fault of course, I couldnt even start the problem lol.

Anyways thanks for the quick reply.