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I've been trying to derive an expression for ∑nCr in the past hour and I got nothing out of it *frown*... I just kept expanding and cancelling the expansion using n!/[r!(n-r)!], I think I'm missing the pattern...
Thanks in advance!
P.S. I couldn't find my high school text book
Last edited by mitochondria (2006-07-03 12:37:11)
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With the summation being on r and n constant, right? I believe it is 2^n.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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Yes Sorry.. I should have mentioned that r = 0
Mm, that sounds familiar Thank you very much.. I'll just go through my algebra and see what went wrong...
Just out of curiosity... n and r can vary =/? (Sorry.. I"m not a mathematician...)
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I should have mentioned that r = 0... Just out of curiosity... n and r can vary =/? (Sorry.. I"m not a mathematician...)
I don't really understand your question, and from them, it appears that you don't have too much experience with summations. So just to make sure I cover all the bases, I'll go through a quick intro to summations, to make sure we are on the same page.
Summation is represented by:
Where i=0 is the starting point, and n is the ending point. You can also replace n with a number:
In which case this sum (for example) would be 0 + 1 + 2 + 3 + 4 + 5 = 15. The shorthand version for summation is:
Which means the summation from 0 to infinity of n. This is very often used in calculus, and so this shorthand was adopted to make things easier to write. Obviously, if you have 2 or more variables, you need to specify which one it is that you are changing. So for example:
Doesn't make any sense while:
Does. Or, you could do a double summation:
But be careful here, as sometimes the order of summation matters.
Hopefully I covered your question somewhere in there.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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Mm... Thank you I do know about summation though, sorry about my rather obscure question. What I was trying to say was that I forgot to specify that the limits are from r=0 to r=n at the beginning, that is:
I went through my algebra again and I just couldn't see how it gives 2^n (which is true for n is odd) *frown*...
Edited: I do know that it's just the summation of a row in the pascal triangle and I can prove it using the bionomial expression (x+y)^n where x = y = 1... I just can't make sense of the algebra and see how it gives 2^n
Last edited by mitochondria (2006-07-03 13:02:14)
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#5 right
(x+y)[sup]n[/sup]= [sub]n[/sub]C[sub]0[/sub]x[sup]n[/sup] +[sub]n[/sub]C[sub]1[/sub]x[sup]n-1[/sup]y +...+[sub]n[/sub]C[sub]n[/sub]y[sup]n[/sup]
let x=1 and y=1
2[sup]n[/sup]= the sum you want
X'(y-Xβ)=0
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What George posted is binomial theorm. It took me a while to understand it even though I knew what he was trying to say! So let me try to post cleaner version:
Binomial Theorm states that:
So if we let x = y = 1:
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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