Competition problem — Combinatorics

Library / /102 of 1394

Combinatorics Difficulty 4.7 AIME Find the answer United States

Problem:
Let $\mathcal{H}$ be the unit hypercube of dimension $4$ with a vertex at $(x, y, z, w)$ for each choice of $x, y, z, w \in \{0,1\}$ . (Note that $\mathcal{H}$ has $2^{4}=16$ 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 $\mathcal{H}$ ) by taking exactly $4$ steps along the edges of $\mathcal{H}$ ?

Your answer

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

Solution

Solution:
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 $24$ .

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