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

How many solutions does the equation

have on the set of positive integers?

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Just use Excel.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 15,152

0, 1, 99

1, 2, 97

2, 3, 95

the list seems endless!

Character is who you are when no one is looking.

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

No, the list is finite. Also, I think *x*, *y*, *z* must be positive, not 0.

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

All right. When *x* = 1, *y* can be any integer from 1 to 98. When *x* = 2, *y* can be any integer from 1 to 97. When *x* = 3, *y* can be any integer from 1 to 96. And so on, until when *x* = 98, *y* can only be 1. Once *x* and *y* have been chosen, the value of *z* is determined. Hence it looks like the total number of solutions is 98 + 97 +
+ 2 + 1 = 4851.

*Last edited by JaneFairfax (2008-01-01 03:29:15)*

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,566

Reminds me of the base of a pyramid with the origin as the apex and

the center of the base near three points (33,33,34), (34,33,33), and (33,34,33).

This base contains all the points needed I declare warily, I caught half of the flu that others got all of it.

**igloo** **myrtilles** **fourmis**

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Here's another way of looking at it:

Think about a number line from 1 to 100. The line has 100 numbers in it, so there are 99 "gaps" between the numbers.

If we place two 'dividers' on the number line, and define x to be the amount left of the dividers, y to be the amount between them and z to be the amount right of them, then tony's question is equivalent to "how many ways can the dividers be placed?"

Clearly, the first divider can be placed in 99 spots and the second can be placed in 98 (99 less the position that the first divider is occupying). Then, the ways to place them are 99x98.

Divide that by two because swapping the two dividers doesn't give a different position, and so the answer is (98x99)/2 = 1+2+...+98.

Why did the vector cross the road?

It wanted to be normal.

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

I like Mathsys solution.

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,566

Cool Neat Awesome!!

Now another way with the tri-pyramid, is to

go on the 3 axes from 0 to 100, which is 101 numbers.

This makes a bowling pin arrangement with 1,2,3,4,5,...101.

Then you strip the axes rows off all three sides which

seems to yield 98 rows, just like Mathsy's answer!!!! 4851 ways.

(I meant strip off the diagonal rows on the outside that connect

between each of the 3 pairs of axes at 100 out on each axis)

So for example, if it were smaller like

vvvv1vvv

vvv2v3vv

vv4v5v6v

v7v8v9vJ, then after stripping off the 1-2-4-7 and the 1-3-6-J and the 7-8-9-J, you

are left with just the 5 in the middle, which is 1 row, or three less than 4 rows you

start with.

*Last edited by John E. Franklin (2008-01-03 07:47:44)*

**igloo** **myrtilles** **fourmis**

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**Identity****Member**- Registered: 2007-04-18
- Posts: 934

Yes nice answer mathsy! This is a really important method for answering questions like:

A toy store owner sells 10 toys in one hour. If he only manufactures 3 types of toy, how many different ways can he sell his 10 toys?

And more advanced questions, such as question 7 here:

http://www.mathscomp.ms.unimelb.edu.au/ … 2007IS.pdf

*Last edited by Identity (2008-01-01 19:50:44)*

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,657

hi gAr

i have seen you use the term GF for a long time.i tried to search it on the net but it didn't clarify anything.could you explain what that is(maybe in another topic)

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

Hi anonimnystefy,

It's kind of intuitive to me. Can't really explain better than the authors.

Are you sure you searched well?

This is a good introduction found in the first page of search results: www.mathdb.org/notes_download/elementar … ae_A11.pdf

Also, there a complete book written about it, available here: http://www.math.upenn.edu/~wilf/DownldGF.html

Aside, how much did you progress with the limit on the other thread?

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,657

thank for the link.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,615

Wilf wrote:

A generating function is a clothesline.

**In mathematics, you don't understand things. You just get used to them.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,657

who is wilf?

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,615

The author of Generatingfunctionology, the first two chapters are about the best there is. After that it stiffens up.

Also it is freely downloadable on the net.

**In mathematics, you don't understand things. You just get used to them.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,657

Ahhhh... How many stupidities have came from me. The answer looks so simple now...

Here lies the reader who will never open this book. He is forever dead.

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