Competition problem — Combinatorics

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Combinatorics Difficulty 4.0 AMC 10/12 Find the answer United States

Problem:

How many distinct sets of 8 positive odd integers sum to 20?

Your answer

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

Solution

Solution:

This is the same as the number of ways 8 nonnegative even integers sum to 12 (we subtract 1 from each integer in the above sum). All 11 possibilities are (leaving out 0s): $12$ , $10+2$ , $8+4$ , $8+2+2$ , $6+6$ , $6+4+2$ , $6+2+2+2+2$ , $4+4+4$ , $4+4+2+2$ , $4+2+2+2+2$ , $2+2+2+2+2+2$ .

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