Grid routes avoiding a closed intersection

Question

You walk a grid going 4 blocks east and 3 blocks north (only east or north moves). Construction has closed the intersection that is 2 blocks east and 2 blocks north of your start, so no route may pass through it. How many shortest routes avoid the closed intersection?

Accepted Answer

Use complementary counting: (all shortest routes) minus (routes through the blocked corner). All routes (4 east, 3 north, 7 steps): \binom{7}{3} = 35. Routes through the blocked corner, split into two legs and multiplied: - Start to corner (2 east, 2 north, 4 steps): \binom{4}{2} = 6. - Corner to finish (2 east, 1 north, 3 steps): \binom{3}{1} = 3. - Through the corner: 6 	imes 3 = 18. Allowed routes: 35 - 18 = 17. Takeaway. To avoid a blocked point, subtract routes forced through it from the total. Here 35 - 18 = 17.

Grid routes avoiding a closed intersection

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