Competition problem — Combinatorics

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Combinatorics Difficulty 4.5 AIME Find the answer United States

Problem:
How many ordered quadruples $(a, b, c, d)$ of four distinct numbers chosen from the set $\{1,2,3, \ldots, 9\}$ satisfy $b<a$ , $b<c$ , and $d<c$ ?

Your answer

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

Solution

Solution:
Given any 4 elements $p<q<r<s$ of $\{1,2, \ldots, 9\}$ , there are 5 ways of rearranging them to satisfy the inequality: prqs, psqr, qspr, qrps, and rspq. This gives a total of $\binom{9}{4} \cdot 5=630$ quadruples.

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