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#1 2010-11-08 09:44:45

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

How many

How many integral solutions does the equation


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#2 2010-11-08 10:24:05

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,248

Re: How many

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#3 2010-11-08 21:16:55

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

Re: How many

Hi bobbym

Try again

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

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#4 2010-11-08 21:29:48

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,248

Re: How many

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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