Math Is Fun Forum

pokemonmaster101

Member

Registered: 2014-09-17

Posts: 5

Generating Functions Problems

1) In how many ways can I collect a total of 20 dollars from 4 different children and 3 different adults, if each child can contribute up to 6 dollars, each adult can give up to 10 dollars, and each individual gives a nonnegative whole number of dollars?

2) Find the sum of the coefficients of

and

in the power series expansion of   

[latex fixed by admin]

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Generating Functions Problems

Hi;

---

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.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Generating Functions Problems

Hi pokemonmaster101

Welcome to the forum!

What's your favourite colour?

Have you set up the GF for the first problem?

Hi bobbym

If your answer to question two is the same for both powers, that is not correct.

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“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
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Generating Functions Problems

hi  pokemonmaster101

Welcome to the forum.

hi Stefy,

I make it the same as bobbym. 

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Generating Functions Problems

Sorry, I should read more carefully. It asked for the sum.

---

“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
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Generating Functions Problems

Tricky thing 'adding zero'. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

pokemonmaster101

Member

Registered: 2014-09-17

Posts: 5

Re: Generating Functions Problems

4) What is the coefficient of x^{11} in the power series expansion of 1/(1-x-x^4)?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Generating Functions 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.

pokemonmaster101

Member

Registered: 2014-09-17

Posts: 5

Re: Generating Functions Problems

5) Determine how many ways I can distribute 80 candies to 3 kids, such that:

The first kid receives an arbitrary number of candies (possibly 0).
The second kid receives an even positive number of candies.
The third kid receives 0, 2, or 5 candies.
Every candy is distributed.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Generating Functions Problems

Hi;

It looks like I am just doing your homework for you.

http://www.mathisfunforum.com/viewtopic.php?id=14654

I have given you the first couple but in order for you to learn you must try to do them yourself. What have you done?
You obviously are in a pretty advanced course where they are throwing generating functions at you what have you learned? What have you tried?

---

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.

pokemonmaster101

Member

Registered: 2014-09-17

Posts: 5

Re: Generating Functions Problems

I have gotten the functions and multiplied them:

(1+x^2+x^5)(1/(1-x)) * (x^2/(1-x^2)) = [x^2 (1+x^2+x^5)]/[(1-x)^2 * (1-x)]= (x^2+x^4+x^7)/(1-x)^3

It looks kinda ugly here so you should write it on paper.  It would look a lot better.

From the other problems, I know that I'm almost done, but I still don't understand how you get past this barrier.

Thank you for your help!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Generating Functions Problems

That sure is ugly and totally impossible to understand. Makes me want to tear my few remaining hairs out ( check out my avatar ).

Please learn to latex up your math.

http://latex.codecogs.com/eqneditor/editor.php

First kids generating function:

Second kid:

Third kid:

Which coefficient do you compute?

---

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.

pokemonmaster101

Member

Registered: 2014-09-17

Posts: 5

Re: Generating Functions Problems

If you multiply them, you get:

And we need to find the x^80 coefficient.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Generating Functions Problems

Hi;

What techniques were you taught to do that?

---

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.