For $0 \lt k \lt n$,
$$\binom{n}{k} = \frac{n}{n-k}\binom{n-1}{k}$$
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]: https://i.sstatic.net/cO2AX.png
Exactly $n-k$ times:
![alt text][2]
[2]: https://i.sstatic.net/N7tTJ.png
By induction, a base case, and taking $k=n$ and $k=0$ for granted: $$\binom{n}{k} = \frac{n}{(n-k)} \frac{(n-1)!}{(n-1-k)!\ k!} = \frac{n!}{(n-k)!\ k!}$$
