Task number: 3452

How many ways are there to walk in Manhattan from the corner of 5th Avenue and 15th Street to the corner of 10th Avenue and 23rd street, assuming that we walk only in a northwestern or northeastern direction?

(In this area streets form a regular grid without shortcuts and with no dead ends or interrupted streets.)

  • Hint

    The walk will be 13 blocks long, of which 5 (longer, about 200m) lead northwest and eight (shorter, about 50m) lead northeast. If from the sequence numbers we choose a five-element set indicating when we will go northwest, we will obtain a unique description of our trip. This also applies in reverse.

  • Answer

    The number of possible paths is \(\binom{13}{5}=1287\).

Difficulty level: Easy task (using definitions and simple reasoning)
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