Does the delivery route pass the coffee shop?

Question

A courier must travel from the southwest corner of a grid of city blocks to the northeast corner that is 4 blocks east and 3 blocks north. She only ever walks east or north (no backtracking), and every such shortest route is equally likely.

A coffee shop sits at the intersection that is 2 blocks east and 1 block north of the start.

What is the probability her route passes directly through that coffee-shop corner?

Accepted Answer

A shortest route is a sequence of 4 easts (E) and 3 norths (N) in some order. The total number of routes is the number of ways to arrange those letters: \binom{7}{3} = 35. A route passes through the coffee shop only if it reaches the point (2 east, 1 north) and then continues to the finish. - Start to coffee shop: arrange 2 E and 1 N, which is \binom{3}{1} = 3 ways. - Coffee shop to finish: the remaining trip needs 2 more E and 2 more N, which is \binom{4}{2} = 6 ways. Routes through the shop: 3 	imes 6 = 18. Therefore P = \frac{18}{35} \approx 0.514. Takeaway. The count of monotone grid pat…

Does the delivery route pass the coffee shop?

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