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.
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
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.
Frequently asked questions
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.