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#26 2014-12-14 05:31:57

gourish
Member
Registered: 2013-05-28
Posts: 153

Re: number of solutions to a linear equation

umm... take x_xxxxxx_xxx_xxx.... so there are groups (x) (xxxxxx) (xxx) (xxx)....  let a=(x) b=(xxxxxx) c=(xxx) d=(xxx)..

the sum of these groups or variables is 13. and i took the c and d as the same as there is no constraint like a,b,c,d are distinct....


"The man was just too bored so he invented maths for fun"
-some wise guy

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#27 2014-12-14 05:34:41

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: number of solutions to a linear equation

Okay, so for your problem we only need to know in the nine spaces

between x x x x x x x x x x how many ways there are to put 2 dashes.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#28 2014-12-14 05:39:08

gourish
Member
Registered: 2013-05-28
Posts: 153

Re: number of solutions to a linear equation

well that's just choosing 2 spaces out of 9.... cool i got it it's (9 chooses 2)... brilliant...


"The man was just too bored so he invented maths for fun"
-some wise guy

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#29 2014-12-14 05:40:51

gourish
Member
Registered: 2013-05-28
Posts: 153

Re: number of solutions to a linear equation

but there's this one problem how did u find that there are 9 spaces when u know that there are 10 x's.... you can put a dash in place of each x.. so there must be 10 spaces right? what  am i missing here....?


"The man was just too bored so he invented maths for fun"
-some wise guy

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#30 2014-12-14 05:42:26

gourish
Member
Registered: 2013-05-28
Posts: 153

Re: number of solutions to a linear equation

oh wait... i can't... big_smile i got it... i can not put a dash at an end... that would mean that one of the groups (variable) is zero which is against my constraint... big_smile


"The man was just too bored so he invented maths for fun"
-some wise guy

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#31 2014-12-14 05:43:18

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: number of solutions to a linear equation

You have to choose two of these dashes:
X_x_x_x_x_x_x_x_x_x


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#32 2014-12-14 05:43:34

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: number of solutions to a linear equation

The generating function does the calculating without doing any of the above reasoning.

x can be any number from 1 to 10 so it is represented by the polynomial

y can be any number from 1 to 10 so it is represented by the polynomial

z can be any number from 1 to 10 so it is represented by the polynomial


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#33 2014-12-14 05:47:29

gourish
Member
Registered: 2013-05-28
Posts: 153

Re: number of solutions to a linear equation

x can be any number from one to ten but why does it mean that it can be represented as x+x^2 and so on till x^10 ?


"The man was just too bored so he invented maths for fun"
-some wise guy

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#34 2014-12-14 05:55:49

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: number of solutions to a linear equation

That is what they call placeholders. The powers of each x represents the number. It is a little complicated and a book is the best place to get a detailed explanation. Anyway, they do the combinatorics for us.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#35 2014-12-14 05:56:38

gourish
Member
Registered: 2013-05-28
Posts: 153

Re: number of solutions to a linear equation

okay.... go on... i have the polynomials representing the variables...


"The man was just too bored so he invented maths for fun"
-some wise guy

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#36 2014-12-14 06:00:29

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: number of solutions to a linear equation

We want to add x, y, z to get 10. So, to add their exponents we multiply the polynomials.

So, we need to multiply

(x+x^2+x^3+...)(x+x^2+x^3+...)(x+x^2+x^3+...)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#37 2014-12-14 06:05:14

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: number of solutions to a linear equation

This is pretty mechanical so you just use the same process basically again and again.

You just multiply the three polynomials:

this equals when expanded.

You now check the coefficient of x^10, what is it?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#38 2014-12-14 06:05:27

gourish
Member
Registered: 2013-05-28
Posts: 153

Re: number of solutions to a linear equation

okay so we now get to back the question that u originally asked me.... right... we now need to find the coefficient of x^10 in this expansion where the whole term is raised to 3....

now getting back to the multinomial theorem....(i have got a fair enough idea of binomial theorem and have read that multinomial theorem is general form of bi theo..)


"The man was just too bored so he invented maths for fun"
-some wise guy

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#39 2014-12-14 06:06:41

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: number of solutions to a linear equation

One thing at a time.

This is pretty mechanical so you just use the same process basically again and again.

You just multiply the three polynomials:

this equals when expanded.

You now check the coefficient of x^10, what is it?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#40 2014-12-14 06:07:59

gourish
Member
Registered: 2013-05-28
Posts: 153

Re: number of solutions to a linear equation

36... the answer...

i bet you didn't expand that term by hand....


"The man was just too bored so he invented maths for fun"
-some wise guy

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#41 2014-12-14 06:08:27

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: number of solutions to a linear equation

We used a computer!


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#42 2014-12-14 06:10:00

gourish
Member
Registered: 2013-05-28
Posts: 153

Re: number of solutions to a linear equation

big_smile so any suggestions for a good book that i can read for combinatorics... including such problems...


"The man was just too bored so he invented maths for fun"
-some wise guy

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#43 2014-12-14 06:11:13

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: number of solutions to a linear equation

The tucker book!


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#44 2014-12-14 06:13:40

gourish
Member
Registered: 2013-05-28
Posts: 153

Re: number of solutions to a linear equation

that's it's name?  "tucker book" on google gave me links to amazon having weird books "they serve beer in hell"  tongue can you be more specific....


"The man was just too bored so he invented maths for fun"
-some wise guy

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#45 2014-12-14 06:14:49

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: number of solutions to a linear equation

i bet you didn't expand that term by hand....

It can be done by hand too and books are full of ways to show you how.

Try this book Applied Combinatorics sixth edition Alan Tucker.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#46 2014-12-14 06:17:54

gourish
Member
Registered: 2013-05-28
Posts: 153

Re: number of solutions to a linear equation

well does it cover the basic combinatorics as well... i think i will need to rebuild my basics to be able to fully appreciate it's applications


"The man was just too bored so he invented maths for fun"
-some wise guy

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#47 2014-12-14 06:20:24

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: number of solutions to a linear equation

Yes, it provides a very strong base for combinatorial reasoning. That is better than books with around 100-200 problems with just 4-5 line solutions without any explanation


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#48 2014-12-14 06:21:09

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: number of solutions to a linear equation

It covers everything but it is tough. Do not read it as you were taught to read a textbook.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#49 2014-12-14 06:22:37

gourish
Member
Registered: 2013-05-28
Posts: 153

Re: number of solutions to a linear equation

ok... well i read that it contains graph theory... why is that necessary for combinatorics?


"The man was just too bored so he invented maths for fun"
-some wise guy

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#50 2014-12-14 06:27:26

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: number of solutions to a linear equation

I do not use it so much myself but it is called by some the replacement for topology or rather the poor man's topology. You have just seen how a combinatorics problem can be turned into an analysis problem when we use generating functions. The ability to change a problem from one branch of math to another is the key to solving more problems.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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