Can someone using only these conditions
$$a_{m,k}=a_{m-1,k}+a_{m-1,k-1},m>k$$
$$a_{m,k}=1,m=k$$
$$a_{m,k}=0,m
prove that
$$a_{m,k}=\frac{m!}{k!(m-k)!}$$
here is way to construct Pascal triangle. I am interested to view direct proof. I read many books on combinatorics and nowhere such proof is done yet.