Competition problem — Combinatorics

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Combinatorics Difficulty 4.5 AIME Find the answer United States

Problem:

How many triples $(A, B, C)$ of positive integers (positive integers are the numbers $1,2,3,4, \ldots$ ) are there such that $A+B+C=10$ , where order does not matter (for instance the triples $(2,3,5)$ and $(3,2,5)$ are considered to be the same triple) and where two of the integers in a triple could be the same (for instance $(3,3,4)$ is a valid triple).

Your answer

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

Solution

Solution:

The triples can merely be enumerated: $(1,1,8)$ , $(1,2,7)$ , $(1,3,6)$ , $(1,4,5)$ , $(2,2,6)$ , $(2,3,5)$ , $(2,4,4)$ , and $(3,3,4)$ . There are 8 elements.

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