# Pascal’s Triangle Generator - Binomial Coefficients and Expansions Online

> Free online Pascal's triangle tool: generate any number of rows, highlight one for the full (a+b)ⁿ expansion and row sum, and look up any C(n, k) exactly.

- URL: https://qilingtools.com/en/math/pascal
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Generate Pascal’s triangle and pick any row to see the matching (a+b)ⁿ expansion and its coefficients. A separate C(n, k) box handles large values of n, and everything is computed with big integers so the numbers stay exact.

## How to use

1. Choose how many rows to display (up to 30)
2. Choose a row to highlight — its coefficients and binomial expansion appear below
3. Use the C(n, k) box for a single binomial coefficient

## Features

- Any number of rows up to 30, copyable as a block
- The highlighted row is expanded in full, with coefficients of 1 omitted as usual
- The row sum 2ⁿ and the largest coefficient in the row
- A standalone C(n, k) lookup that handles n up to 1000
- BigInt recursion throughout, so even huge entries are exact
- Notes on the diagonals: natural numbers, triangular numbers, tetrahedral numbers and Fibonacci

## FAQ

### How is Pascal’s triangle built?

Each row begins and ends with 1, and every entry in between is the sum of the two entries above it — the identity C(n,k) = C(n−1,k−1) + C(n−1,k).

### How does it relate to the binomial theorem?

Row n holds the coefficients of (a+b)ⁿ. Row 5 is 1 5 10 10 5 1, which gives a⁵+5a⁴b+10a³b²+10a²b³+5ab⁴+b⁵.

### What does each row add up to?

Row n sums to 2ⁿ (put a = b = 1 into (a+b)ⁿ). Adding with alternating signs gives 0 instead (a = 1, b = −1).

### What patterns are in the diagonals?

The first diagonal is all ones, the second the natural numbers, the third the triangular numbers 1, 3, 6, 10…, the fourth the tetrahedral numbers. Sums along the shallow diagonals produce the Fibonacci sequence.

### Why is it also called Yang Hui’s triangle?

The Chinese mathematician Yang Hui recorded it in 1261, crediting the earlier Jia Xian. In the West it carries Pascal’s name after his systematic study of it in 1654.

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