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 pseudocode, the nchooser 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  (i1)
  
  i
i=1
japhy 
Perl and Regex Hacker
Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
Read Where should I post X? if you're not absolutely sure you're posting in the right place.
Please read these before you post! —
Posts may use any of the Perl Monks Approved HTML tags:
 a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)

For: 

Use: 
 &   & 
 <   < 
 >   > 
 [   [ 
 ]   ] 
Link using PerlMonks shortcuts! What shortcuts can I use for linking?
See Writeup Formatting Tips and other pages linked from there for more info.