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Problem 1052

AMC 12 late, AIME early

Combinatorics Difficulty 5.0 Prove it $12^{\text {th }}$ Annual Harvard-MIT Mathematics Tournament · United States

How many ways can the integers from $-7$ to $7$ be arranged in a sequence such that the absolute value of the numbers in the sequence is nondecreasing?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Solution:

Each of the pairs $a, -a$ must occur in increasing order of $a$ for $a = 1, \ldots, 7$ , but $a$ can either occur before or after $-a$ , for a total of $2^{7} = 128$ possible sequences.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.