Competition problem — Combinatorics

Combinatorics Difficulty 5.0 AIME Prove it United States

Problem:

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?

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.

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