HMMT 二月 2015 · 冲刺赛 · 第 5 题

HMMT February 2015 — Guts Round — Problem 5

[ 5 ] Let H be the unit hypercube of dimension 4 with a vertex at ( x, y, z, w ) for each choice of x, y, z, w ∈ 4 { 0 , 1 } . (Note that H has 2 = 16 vertices.) A bug starts at the vertex (0 , 0 , 0 , 0). In how many ways can the bug move to (1 , 1 , 1 , 1) (the opposite corner of H ) by taking exactly 4 steps along the edges of H ?

解析

[ 5 ] Let H be the unit hypercube of dimension 4 with a vertex at ( x, y, z, w ) for each choice of x, y, z, w ∈ 4 { 0 , 1 } . (Note that H has 2 = 16 vertices.) A bug starts at the vertex (0 , 0 , 0 , 0). In how many ways can the bug move to (1 , 1 , 1 , 1) (the opposite corner of H ) by taking exactly 4 steps along the edges of H ? Answer: 24 You may think of this as sequentially adding 1 to each coordinate of (0 , 0 , 0 , 0). There are 4 ways to choose the first coordinate, 3 ways to choose the second, and 2 ways to choose the third. The product is 24.