HMMT 二月 2015 · 冲刺赛 · 第 1 题

HMMT February 2015 — Guts Round — Problem 1

[ 4 ] 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. ◦

解析

[ 4 ] 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. 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. ◦