Pascal Triangle Calculator
This Pascal triangle calculator builds the triangle up to row 60: every number is the sum of the two above it, and row n gives the coefficients of (a + b)ⁿ. For n = 6 it is 1 6 15 20 15 6 1 – shown with the written-out binomial expansion and the main properties.
Example: Row n 6 · Binomial expansion: (a + b)ⁿ → Row n: 1 6 15 20 15 6 1. Source: Wikipedia – Pascalsches Dreieck. Updated: .
How it is calculated
Pascal’s triangle is an arrangement of numbers in which each number is the sum of the two directly above it, with 1s down the edges. Row 0 is 1, row 1 is 1 1, row 2 is 1 2 1, row 3 is 1 3 3 1 and so on. The entries are the binomial coefficients “n choose k”.
What can you do with it?
- Binomial expansion: Row n holds the coefficients of (a + b)ⁿ. From 1 3 3 1 you get (a + b)³ = a³ + 3a²b + 3ab² + b³. For (a − b)ⁿ the signs alternate.
- Row sums: Row n adds up to 2ⁿ – for example 1 + 4 + 6 + 4 + 1 = 16 = 2⁴.
- Probability: “n choose k” counts the ways to pick k of n things, as in coin tosses or lottery combinations.
- Fibonacci numbers: Add the shallow diagonals and you get 1, 1, 2, 3, 5, 8, 13 …
How the tool works
The triangle is built row by row by addition, using exact whole numbers – up to row 60 with values above 1017, without rounding errors. For single coefficients, permutations and variations use the combination calculator; more sequences are in the sequence calculator.
Frequently asked questions
What is Pascal’s triangle?
A triangular arrangement of numbers in which each number is the sum of the two above it. It contains all binomial coefficients and is named after Blaise Pascal, though it was known earlier.
How do you work out a row of Pascal’s triangle?
Start with 1 and in each next row add neighbouring numbers of the row above; the edges are 1 again. Row 5 is 1 5 10 10 5 1.
How does Pascal’s triangle relate to binomial expansion?
Row n gives the coefficients of (a + b)ⁿ. For n = 2, 1 2 1 gives the familiar a² + 2ab + b².
Where is the Fibonacci sequence in the triangle?
In the shallow diagonals: their sums are 1, 1, 2, 3, 5, 8, 13 and so on. The tool shows the Fibonacci number that matches the chosen row.
What is the sum of a row?
The sum of row n is always 2 to the power n. Row 10 adds up to 1,024.
Sources and legal basis
- Wikipedia – Pascalsches Dreieck
- Wikipedia – Pascal's triangle
- OEIS A007318 – Pascal's triangle read by rows
As of: · Editorial policy
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