# Pascal’s Triangle Calculator

URL: https://speedycalculator.com/calculators/math/pascals-triangle/
Last updated: 2026-10-30

> The Pascal's triangle calculator shows any row of Pascal's triangle (the binomial coefficients C(n, k)), its sum and middle value, and draws the triangle up to that row.

## Inputs

- Row n

## Results

- Row n
- Sum of the row (2ⁿ)
- Largest (middle) value
- Pascal's triangle

## Formula

```
Row n: C(n, 0), C(n, 1), …, C(n, n),  C(n, k) = C(n − 1, k − 1) + C(n − 1, k)
Row sum = 2ⁿ
```

## Explanation

Each number in Pascal's triangle is the sum of the two above it. Its rows give the coefficients of (a + b)ⁿ, counts of combinations, probabilities of coin flips, and hide the Fibonacci numbers, powers of 11 and the Sierpiński triangle.

### How to use the Pascal's Triangle Calculator

- **Row n**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Enter the row number n (the top row is row 0). Values are exact whole numbers.

## Worked example

Inputs — Row n: 6.

Results — **Row n = 1, 6, 15, 20, 15, 6, 1**; Sum of the row (2ⁿ) = 64; Largest (middle) value = 20; Pascal's triangle = 1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1.

## Key terms

- **Binomial coefficient**: C(n, k), the number of ways to choose k items from n.

## FAQ

**What is row 4?**
1, 4, 6, 4, 1 – the coefficients of (a + b)⁴.

**Why do rows sum to powers of 2?**
Each row counts all subsets of an n-element set.

**Where are the Fibonacci numbers?**
In the sums of shallow diagonals.

**What happens if I colour the odd numbers?**
You get the Sierpiński triangle.


## Sources

- [Wolfram MathWorld – Pascal's triangle](https://mathworld.wolfram.com/PascalsTriangle.html)