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

Hi bobbym,

Ok, I understand! Thanks.

Can we extend it to more than 2 players?

"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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,352

Hi gAr;

I do not think so. We can have the positive exponents represent one player and the negative for the other. There just is not any other choices.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

Hi bobbym,

Ok, thank you.

I too could not find a way.

"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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,352

I guess if you had 3 you might be able to do the problem 2 at a time.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

Or can we have 3 different variables in the generating functions, and then add the coefficients as we require?

"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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,352

Yes, we could. But that would get complicated. The beauty of the powers and the reciprocals is that they cancel out automatically.

Take this one for example:

This says that B loses by exactly 11 strokes 1975 / 308367 percent of the time. We do not have to think about the answer, the GF's do it 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.****No great discovery was ever made without a bold guess. **

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

Yes, I too like it.

By the way, nice problem!

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,352

Thanks glad you looked at it.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,352

New Problem!

**D poses this one. A guy has 5 bananas, 5 pears, 5 apples and 5 oranges. How many different arrangements of 4 groups can he make?**

A says) Using Sylow p subgroup I come up with 1460.

B says) I remember them but I do not see how they apply here. Anyway that answer is wrong. The correct answer is...

C says) Hold on B, do not just blurt it out, I am still working on it.

D says) Hold it guys, I got hungry and ate the fruit. Now there are 0 ways to arrange the fruit.

Ignoring D's answer what do you get?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

Hi bobbym,

Did not see your post.

How many different arrangements of 4 groups can he make?

Does this mean arrangements in groups of 4?

We can use exponential generating function which you taught me!

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,352

Hi gAr;

Very good! Yes, wanted to see if you remembered that.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

Hi bobbym,

Thanks.

Anyway, the

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,352

Kayrect!

Generating Functions,Generating Functions,Generating Functions,Generating Functions,Generating Functions,Generating Functions,Generating Functions,Generating Functions!!!!

Hip hip hooray

Use them

And all our combinatorics problems go away.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

Yep, they are sooooo coooooooooooool!!!

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,352

New Problem!

C says if two people play russian roulette they have an even chance to die. He explains the rules that 1 bullet is put into the cylinder that can hold six shells. They spin the cylinder each time before firing. They take turns pointing the gun at there own head and pulling the trigger! They are obviously imbeciles.

A says) Sounds reasonable to me. Not even much of a problem to pose. Yes they have an even chance.

B says) Nope, the first player, meaning the first guy to point the gun at his own head is a big underdog.

C says) I posed it and they have an even chance.

D says) If I have to play I want to go first you know to get it out of the way.

How big is the second players advantage?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

Hi bobbym,

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

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

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,352

Hi gAr;

That is correct! Very good! It seems there is always another way.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

Hi bobbym,

Yes.

These are the only methods I know so far!

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,352

Hi;

That is all I can think of too.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

Hi bobbym,

I thought of a variation of your question: (Actually, this is how I misunderstood your question before reading it again!)

There are 2 players with one pistol each.

The cylinders are fully loaded in the beginning.

Both of them have an accuracy of 50 percent.

They spin the cylinder each time before firing. They take turns to shoot the other person. ( After pulling the trigger, other player gets the chance )

i) What is the probability that second player survives?

ii) What is the probability that both survive?

I thought markov chain would be easy here too, but it's not!

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,352

Hi gAr;

Markov chains have some problems without replacement and this is what your problem is. For both to have a chance to live they must run out of bullets.

I solved it like this:

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

Hi bobbym,

Thanks for the solution!

You know what, the markov chain which I thought was incorrect gave the same answer!

Three cheers to Markov Chain!

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,352

Hi gAr;

That is two ways then. Thanks for working on it.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

Your answer helped me to verify, thanks.

Though, I'm not fully convinced with the transitions in the chain!

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

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