How many solutions

tony123 · 2007-12-31 22:10:07

How many solutions does the equation

have on the set of positive integers?

JaneFairfax · 2008-01-01 01:54:05

Just use Excel.

Jai Ganesh · 2008-01-01 02:31:02

0, 1, 99
1, 2, 97
2, 3, 95
the list seems endless!

JaneFairfax · 2008-01-01 03:08:46

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

JaneFairfax · 2008-01-01 03:26:52

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)

John E. Franklin · 2008-01-01 03:30:07

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.

mathsyperson · 2008-01-01 03:58:04

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.

JaneFairfax · 2008-01-01 06:06:57

I like Mathsys solution.

John E. Franklin · 2008-01-01 08:29:55

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)

Identity · 2008-01-01 19:48:16

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)

gAr · 2011-07-28 02:37:37

anonimnystefy · 2011-07-28 02:41:59

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)

gAr · 2011-07-28 02:54:27

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?

anonimnystefy · 2011-07-28 02:58:52

thank for the link.

bobbym · 2011-07-28 03:00:56

Wilf wrote:

A generating function is a clothesline.

anonimnystefy · 2011-07-28 03:10:18

who is wilf?

bobbym · 2011-07-28 03:12:57

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.

anonimnystefy · 2012-05-02 11:32:35

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

Math Is Fun Forum

#1 2007-12-31 22:10:07

How many solutions

#2 2008-01-01 01:54:05

Re: How many solutions

#3 2008-01-01 02:31:02

Re: How many solutions

#4 2008-01-01 03:08:46

Re: How many solutions

#5 2008-01-01 03:26:52

Re: How many solutions

#6 2008-01-01 03:30:07

Re: How many solutions

#7 2008-01-01 03:58:04

Re: How many solutions

#8 2008-01-01 06:06:57

Re: How many solutions

#9 2008-01-01 08:29:55

Re: How many solutions

#10 2008-01-01 19:48:16

Re: How many solutions

#11 2011-07-28 02:37:37

Re: How many solutions

#12 2011-07-28 02:41:59

Re: How many solutions

#13 2011-07-28 02:54:27

Re: How many solutions

#14 2011-07-28 02:58:52

Re: How many solutions

#15 2011-07-28 03:00:56

Re: How many solutions

#16 2011-07-28 03:10:18

Re: How many solutions

#17 2011-07-28 03:12:57

Re: How many solutions

#18 2012-05-02 11:32:35

Re: How many solutions

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