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**tony123****Member**- Registered: 2007-08-03
- Posts: 189

How many integral solutions does the equation

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,295

Hi tony123;

To answer this you use generating functions. There are other ways but this is my favorite.

This can be modelled as the polynomial expansion of

with 0<= a,b,c,<=9

The coefficient of x^9 is the answer. We see that it is 55. So there are 55 solutions.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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**tony123****Member**- Registered: 2007-08-03
- Posts: 189

Hi bobbym

Try again

|-5|+|-4|+|0|=9

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 104,295

Hi tony123;

Sorry for the slip up in the other post!

This is the new GF. Checking the coefficient of x^9 we see that it is 326. That is the answer.

Here is another way to do it. One that has the weight of authority behind it, The following formula was developed by Polya and Szego in 1926.

The number of solutions of:

is:

So to get p=9 we subtract the two sums:

Same answer as above.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **A number by itself is useful, but it is far more useful to know how accurate or certain that number is.**

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