Competition problem — Combinatorics

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Combinatorics Difficulty 3.9 AMC 10/12 Find the answer Philippines

Problem:

How many ordered 5-tuples $(a, b, c, d, e)$ of integers satisfy $10 < a < b < c < d < e < 20$ ?

Your answer

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

Solution

Solution:

We simply need to choose which five of the integers $11, 12, \ldots, 19$ will comprise the 5-tuple. Once the integers are chosen, there is only one way to assign them as $a, b, c, d, e$ to satisfy the given inequalities. Thus, there are $\binom{9}{5} = 126$ possible 5-tuples.

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