HMMT 二月 2015 · 冲刺赛 · 第 5 题
HMMT February 2015 — Guts Round — Problem 5
题目详情
英文原题
- [ 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 ∈
{ 0 , 1 } . (Note that H has 2 = 16 vertices.) A bug starts at the vertex (0 , 0 , 0 , 0). In how many ways 4
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
英文解析
- [ 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 ∈
{ 0 , 1 } . (Note that H has 2 = 16 vertices.) A bug starts at the vertex (0 , 0 , 0 , 0). In how many ways 4
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). Thereare 4 ways to choose the first coordinate, 3 ways to choose the second, and 2 ways to choose the third.
The product is 24.