Maths Olympiad Prep

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HMMT February

Combinatorics Difficulty 4.5 AIME Find the answer United States

Problem:
Let RR be the rectangle in the Cartesian plane with vertices at (0,0)(0,0), (2,0)(2,0), (2,1)(2,1), and (0,1)(0,1). RR can be divided into two unit squares, as shown.
Figure 1
The resulting figure has 7 segments of unit length, connecting neighboring lattice points (those lying on or inside RR). Compute the number of paths from (0,1)(0,1) (the upper left corner) to (2,0)(2,0) (the lower right corner) along these 7 segments, where each segment can be used at most once.

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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