Rook routes on a five-by-two block walk

Question

A delivery robot must travel 5 blocks east and 2 blocks north across a grid of streets, moving only east or north. How many different shortest routes can it take?

Accepted Answer

A shortest route is an ordering of 5 east-steps and 2 north-steps, 7 steps in all. Pick which 2 of the 7 slots are north-steps; the rest are east: \binom{7}{2} = \frac{7 	imes 6}{2 	imes 1} = 21. Choosing the 5 east-steps gives the same value, \binom{7}{5} = 21. Takeaway. With a east and b north steps, the count is \binom{a+b}{b}. Here \binom{7}{2} = 21.

Rook routes on a five-by-two block walk

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