in reply to
Re: Binomial Expansion

in thread Binomial Expansion

Ok, for fun, and because I like mathematical proofs (I made it to the USA Mathematic Olympiad twice), here's the proof I use.
In pseudo-code, the n-choose-r function looks like:

`nCr (n,r) {
res = 1;
for i in 1 .. r {
res *= n;
n--;
res /= i;
}
return res;
}
`

The question is, why does

`res` remain an integer? It is not difficult to show that

`res /= 1` is an integer, but how can we prove that by the time we get to

`res /= 7`,

`res` is a multiple of 7?

The proof is that by the time we have gotten to `res /= `*X*, we have multiplied `res`'s original value, 1, by *X* continuous integer, and in every series of *X* continuous integers, there will be one number divisible by *X*:

`X = 2, series = (y, y+1); one is div. by 2
X = 3, series = (y, y+1, y+2); one is div. by 3
...
in nCr(17,7),
n is of the series (17,16,15,14,13,12,11)
i is of the series ( 1, 2, 3, 4, 5, 6, 7)
(17/1)
(17/1) (16/2)
(17/1) (16/2) (15/3)
(17/1) (16/4) (15/3) (14/2)
(17/1) (16/4) (15/(3*5)) (14/2) (13/1)
(17/1) (16/4) (15/(3*5)) (14/2) (13/1) (12/6)
(17/1) (16/4) (15/(3*5)) (14/(2*7)) (13/1) (12/6) (11/1)
`

We're allowed to move the denominators around like I did, because we've already showed the product will already be an integer. The product looks like:

` r
-----
| | n - (i-1)
| | ---------
| | i
i=1
`

`japhy` --

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