From the wikepedia,
For self-avoiding walks from one end of a diagonal to the other, with only moves in the positive direction, there are exactly $$ \binom{n+m}{n,m} $$paths for an $m × n$ rectangular lattice.
Can I write $\binom{n+m}{n,m}$ = $\frac{(n+m)!}{n! \times m!}$? If not, how to evaluate $\binom{n+m}{n,m}$?
This is just the binomial coefficient and is usually written as: $$\binom {n+m}n=\binom {n+m}m=\frac {(n+m)!}{n!m!}$$
NB: an example of a multinomial coefficient would be $$\binom {n+m+p+q}{n,m,p,q}=\frac {(n+m+p+q)!}{n!m!p!q!}$$