Pascal’s Triangle Calculator
Get any row of Pascal’s triangle up to row 60 with exact big numbers, or print the triangle itself, plus the row sum 2ⁿ, the middle entry and the binomial coefficients C(n, k).
How it works
Entry k of row n = C(n, k) = n! / (k!(n − k)!) · each entry = sum of the two above it: C(n, k) = C(n − 1, k − 1) + C(n − 1, k) · row sum = 2ⁿ
Pascal’s triangle starts with a 1 in row 0; every other number is the sum of the two numbers directly above it. Row n lists the binomial coefficients C(n, 0) … C(n, n), which are the coefficients of (a + b)ⁿ and count the ways to choose k items from n. The rows are symmetric, sum to 2ⁿ and are largest in the middle. The calculator uses exact integer arithmetic, so even row 60 has no rounding.
Worked example
Row 6 is 1, 6, 15, 20, 15, 6, 1: for example 15 = 5 + 10 and 20 = 10 + 10 from row 5 (1, 5, 10, 10, 5, 1). It has 7 entries, sums to 2⁶ = 64, and its middle entry C(6, 3) = 20 is the largest. So (a + b)⁶ = a⁶ + 6a⁵b + 15a⁴b² + 20a³b³ + 15a²b⁴ + 6ab⁵ + b⁶.
Sources: Wolfram MathWorld — Pascal’s Triangle · OpenStax College Algebra 2e — 9.6 Binomial Theorem
Frequently asked questions
How do you find a row of Pascal’s triangle without drawing it?
Use C(n, k) = n!/(k!(n − k)!), or build the row left to right: start at 1 and multiply by (n − k + 1)/k for each next entry. Row 6: 1, 6, 6·5/2 = 15, 15·4/3 = 20, …
Is the top row 0 or 1?
By convention the single 1 at the top is row 0, so row n has n + 1 entries and its second entry is n.
What is Pascal’s triangle used for?
Binomial expansions (a + b)ⁿ, counting combinations, coin-flip probabilities (row n ÷ 2ⁿ) and patterns such as triangular numbers, powers of 11 and the Fibonacci numbers along shallow diagonals.
Why does each row add up to a power of 2?
Setting a = b = 1 in (a + b)ⁿ gives 2ⁿ, and it also counts all subsets of an n-element set.
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This calculator is for educational purposes. Double-check important results before acting on them.