# N Choose 2 is the Sum of the First N-1 Integers

DevFeed: [N Choose 2 is the Sum of the First N-1 Integers](<https://devfeed.tech/articles/n-choose-2-is-the-sum-of-the-first-n-1-integers-40243.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2011/10/02/n-choose-2/>)

Published: 2011-10-02T16:16:47Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Mathematics](<https://devfeed.tech/topics/mathematics.md>)

Tags: [arithmetic](<https://devfeed.tech/tags/arithmetic.md>), [bijections](<https://devfeed.tech/tags/bijections.md>), [combinatorics](<https://devfeed.tech/tags/combinatorics.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [proofs-without-words](<https://devfeed.tech/tags/proofs-without-words.md>)

## AI overview

A combinatorial bijection shows that the binomial coefficient n choose 2 equals the sum of the first n−1 integers. The article explains how yellow dots correspond uniquely to pairs of dots in the bottom row, then briefly connects bijections to isomorphism and classification in mathematics.

## Source excerpt

Problem: Determine an arithmetic expression for $ \binom{n}{2}$. Solution: The following picture describes a bijection between the set of yellow dots and the set of pairs of purple dots: In particular, selecting any yellow dots and travelling downward along diagonals gives a unique pair of blue dots. Conversely, picking any pair of blue dots gives a unique yellow dot which is the meeting point (the "peak") of the inward diagonals. If we say the bottom row has $ n$ elements, then the number of yellow dots is clearly $ 1 + 2 + \dots + (n-1)$, and the number of pairs in the last row is just $ \binom{n}{2}$.