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#1 2007-03-04 05:36:43

Christine Y
Guest

Why does Pascal's Pyramid work?

http://www.freewebs.com/roslynstudents2011/Pascal%20rough2.pdf (this is my paper)

I have a math presentation in less than a week, and I have been working hard to figure out why it works. I thought it was something to do with (a+b+c)^2 multiplied by another (a+b+c) is (a+b+c)^3. But I just can't understand why that if you add the three numbers in the lower layer, you get a number in the higher layer.

Also, in my paper, I have nothing that deals with combination. I saw that it is used with the Pascal's triangle, but does it with Pascal's pyramid? And if my paper has no formula that deals with the layers other than (a+b+c)^n, do I have to redo my paper?...

Please help... for my brain is not good enough to do this on my own.:(

#2 2007-03-04 06:04:30

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Why does Pascal's Pyramid work?

If you enclose hyperlinks in URL tags you can make your them clickable, like this:

http://www.freewebs.com/roslynstudents2 … rough2.pdf

Unfortunately I’m unable to access your webpage. I get a message saying that my security settings do not allow the file to be downloaded. sad Is your Pascal’s pyramid something like this:

http://en.wikipedia.org/wiki/Pascal%27s_pyramid

Last edited by JaneFairfax (2007-03-04 09:13:53)

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#3 2007-03-04 08:41:18

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,711

Re: Why does Pascal's Pyramid work?

Also: Pascal's Triangle (the "2D" version of the pyramid)

It works in factorising, because the addition as you go down the pyramid matches the way you add terms.

In the simplest case (x+a)[sup]2[/sup] = (x+a)(x+a) you multiply x by x, x by a, a by x and a by a. Because x by a = a by x, you get 2 like terms, which is the same as:

  1  1
1  2  1

You need to generalise from there, I guess smile


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#4 2007-03-04 09:28:09

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Why does Pascal's Pyramid work?

Well, the 2D case works because

holds for all non-negative integers n and r with rn – and the

’s are the coefficients of the terms in in the binomial expansion of (x+y)[sup]n[/sup].

For the 3D pyramid, I suppose you just need to find a corresponding formula for the coefficients of the terms in the trinomial expansion of (x+y+z)[sup]n[/sup].

Last edited by JaneFairfax (2007-03-04 09:29:41)

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#5 2007-03-05 03:23:09

Patrick
Real Member
Registered: 2006-02-24
Posts: 1,005

Re: Why does Pascal's Pyramid work?

Just a little comment on the paper:

Pascal’s Pyramid is not actually made up by Pascal, but named this way because is has similar properties.

It has similar properties to Pascal? You mean Pascal's triangle by any chance? wink I know it's not a very critical mistake, a mistake none the less smile


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#6 2007-03-05 08:44:16

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Why does Pascal's Pyramid work?

Incidentally, Pascal's triangle wasn't invented by him either, it's just named that way because he did a lot of research on it.


Why did the vector cross the road?
It wanted to be normal.

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#7 2007-03-05 08:48:28

lightning
Real Member
Registered: 2007-02-26
Posts: 2,060

Re: Why does Pascal's Pyramid work?

yep


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#8 2007-03-05 23:51:47

George,Y
Member
Registered: 2006-03-12
Posts: 1,379

Re: Why does Pascal's Pyramid work?

Just combination.


X'(y-Xβ)=0

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#9 2007-03-06 09:08:38

lightning
Real Member
Registered: 2007-02-26
Posts: 2,060

Re: Why does Pascal's Pyramid work?

yep nice avatar


Zappzter - New IM app! Unsure of which room to join? "ZNU" is made to help new users. c:

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