Competition problem — Combinatorics

Library / /12 of 1394

Combinatorics Difficulty 4.5 AIME Find the answer United States

Problem:
Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0)$ , $(2,0)$ , $(2,1)$ , and $(0,1)$ . $R$ can be divided into two unit squares, as shown.

The resulting figure has 7 segments of unit length, connecting neighboring lattice points (those lying on or inside $R$ ). Compute the number of paths from $(0,1)$ (the upper left corner) to $(2,0)$ (the lower right corner) along these 7 segments, where each segment can be used at most once.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
Answer: 4

Just count them directly. If the first step is to the right, there are 2 paths. If the first step is downwards (so the next step must be to the right), there are again 2 paths. This gives a total of 4 paths.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.