Paths across a six-by-six grid

Question

A grid of 6 by 6 squares is drawn on paper (so it has 7 by 7 grid points). You start at the top-left corner point and walk along the grid lines to the bottom-right corner point, moving only right or down, one grid unit at a time. How many different routes are there? Explain the counting idea, and say how the answer changes if the grid is 6 squares wide but only 4 squares tall.

Accepted Answer

924 routes. Every route has the same length. To get from the top-left corner to the bottom-right corner you must move right exactly 6 times and down exactly 6 times, in some order. So every route is a sequence of 12 moves containing six Rs and six Ds. Counting the sequences. A route is determined by choosing which 6 of the 12 positions in the sequence are the down moves (the rest are rights). The number of ways to choose 6 positions out of 12 is 12 choose 6 = 12! / (6! × 6!) = 924. A 6 by 4 grid. Now you need 6 rights and 4 downs, 10 moves in all, and you choose which 4 are downs: 10 choose 4…

Paths across a six-by-six grid

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