Rook routes and binomial coefficients

Count the shortest routes for a piece crossing a grid, moving only right and up one square at a time. The count at a square is the sum of the counts to its left and below, so the grid fills with Pascal's triangle, now larger. The number in the square $f$ right and $r$ up is the binomial coefficient $\binom{f+r}{f}$.

Reading a value directly is choosing which steps go right: a route with $f$ right-steps and $r$ up-steps is a sequence, and choosing the positions of the right-steps among $f+r$ steps gives $\binom{f+r}{f}$ routes. Counting and choosing are the same act.