HMMT February

Combinatorics Difficulty 4.9 AIME Find the answer

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

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, $q s p r, q r p s$ , and $r s p q$ . This gives a total of $\binom{9}{4} \cdot 5=630$ quadruples.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.