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**ElainaVW****Member**- Registered: 2013-04-29
- Posts: 291

Hello;

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,518

Hi Nehushtan

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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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,378

hi Nehushtan,

Using standard calculus:

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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The problem was meant to be easy in other words, use AMGM rather than calculus:

Next problem (more challenging this time):

*Last edited by Nehushtan (2013-08-25 19:31:01)*

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

Hi Nehushtan,

*Last edited by gAr (2013-08-31 19:48:40)*

"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: 15,518

Hi Nehushtan

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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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,518

Yes, but I liked this method, so I posted that. And besides, induction would require a bit of latexing, and I'm on my phone.

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: 86,242

Hi;

We can do the problem directly using the summation calculus.

The first 2 falling factorials of k are

So obviously

The summation calculus can now sum that using the rule

where (r+1) and (r) are the falling factorial operator. Notice the above rule is the discrete counterpart to the integral operator.

Now we finish up with

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**ElainaVW****Member**- Registered: 2013-04-29
- Posts: 291

Hello:

The best way is to work like the M would, experimentally! Although he does seem to have borrowed the idea from Doctor Z as he calls him.

Form a difference table:

{3,6,10,15,21,28,36,45,55}

{3,4,5,6,7,8,9,10}

{1,1,1,1,1,1,1}

The third row is constant so it is a cubic

Using bobbym's idea in

http://www.mathisfunforum.com/viewtopic … 18#p285718

which is the binomial in the question. You should use induction to show that the formula is correct.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,242

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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

By generating functions, start with the g.f of the triangular numbers:

"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: 15,518

gAr wrote:

By generating functions, start with the g.f of the triangular numbers:

That also proves the equality from #10.

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

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

That's right.

"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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Jim loves to play online games. On the online games site he has just joined, he needs a four-digit secret combination code for access. Being very bad at memorizing combinations, even four-digit ones, he writes down his combination code and proceeds to memorize it.

Along comes Jims nosy little brother. As Jims attention is momentarily engaged elsewhere, his little brother sees the piece of paper on which the combination code, picks it up, and looks at it. But Jim, immediately noticing what his little brother is doing, snatches the piece of paper away from him. Go away! he shouts at his little brother. This is private.

Jims little brother meekly trudges away, thinking to himself: I dont remember the four-digit number I just saw, but Ive noticed something. All the digits are different, and the first digit on the left is an odd number.

Meanwhile Jim is worried that his little brother has seen his combination code and might use it to access his account on the online games site. He decides to change the code. Being a bad memorizer of numbers, he simply swaps two of the digits of his existing code to make a new four-digit code. To help him remember which two digits he has swapped, he makes the following notes: (a) subtracting his new code from his old code gives 891, and (b) the sum of the two digits he did not swap is 15.

Good, says Jim, crumpling up the paper on which his code has been written. Now I shall be able to remember my new code from my old one.

What was Jims old four-digit code, and what is his new one?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,242

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,242

Hi;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,242

Hi;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,242

Hi;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**