For $0 \le k \lt n$,
$$\binom{n}{k} = \frac{n}{n-k}\binom{n-1}{k}$$
How k-subsets of [n], marked dark green in the rows, come from k-subsets of [n-1] after n-fold duplication and rearrangement:
![alt text][1] [1]: http://i43.tinypic.com/37rcn.jpg
Exactly $n-k$ times:
![alt text][2]
[2]: http://i44.tinypic.com/2nsozk1.jpg
By induction, a base case, and case $k=n$ which we already know due to case $k=0$: $$\binom{n}{k} = \frac{n}{(n-k)} \frac{(n-1)!}{(n-1-k)!\ k!} = \frac{n!}{(n-k)!\ k!}$$