Counting routes with Pascal

A counter starts at the bottom-left and moves only up or right. How many shortest routes reach each square? A route to a square arrives from the square below or the square to the left, so the count at a square is the sum of those two neighbours. Fill the edge squares with $1$, since there is one way along an edge, and add inward.

The numbers are the binomial coefficients: the count at the square $f$ right and $r$ up is $\binom{f+r}{f}$. Laid out this way they are exactly Pascal's triangle. Additive counting turns a hard enumeration into simple sums.