Right-and-down paths across a 6x6 grid and N by M

A game character occupies the top-left square $(1,1)$ of a $6\times6$ grid and may only step right or down.

(a) How many routes take it to the bottom-right square $(6,6)$?

(b) Describe an algorithm that counts such routes on a general $N\times M$ grid.

(a) $252$. Every route uses exactly $5$ right and $5$ down steps, so it is fixed by choosing which $5$ of the $10$ steps go right: $\binom{10}{5}=252$. (b) Either return $\binom{N+M-2}{N-1}$ directly, or fill a DP table with $P(i,j)=P(i-1,j)+P(i,j-1)$ and ones along the first row and column, which is $O(NM)$ time and $O(\min(N,M))$ memory with a rolling row.