Pascal's Triangle Calculator

Compute any row of Pascal's Triangle, find specific binomial coefficients C(n,k), display the first N rows, and explore identities: Hockey Stick, Vandermonde, Fibonacci diagonals, and Sierpinski triangle.

Row n values
—
Sum of row (= 2ⁿ) —
Alternating sum (= 0 for n≥1) —
Extended More scenarios, charts & detailed breakdown ▾
Row coefficients
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Sum = 2ⁿ —
Fibonacci diagonal sum —
Professional Full parameters & maximum detail ▾

Row

Row n —

Identities & Connections

Hockey Stick Identity —
Vandermonde's Identity (note) —
Sierpinski Triangle (mod 2) —
Central Binomial & Catalan —

How to Use This Calculator

  1. Enter row n (0-indexed) to see all binomial coefficients in that row.
  2. Use First N Rows tab to display the triangle up to N rows.
  3. Use C(n,k) Entry tab to find a specific binomial coefficient.
  4. Use Professional for identities, Sierpinski pattern, and Catalan connection.

Formula

C(n,k) = n! / (k!(n−k)!)

Row sum = 2ⁿ | Alternating sum = 0 (n≥1) | C(n,k) = C(n−1,k−1) + C(n−1,k)

Example

Row 6: [1, 6, 15, 20, 15, 6, 1]. Sum=2⁶=64. C(6,3)=20 (ways to choose 3 items from 6). Alternating sum=1−6+15−20+15−6+1=0.

Frequently Asked Questions

  • Start with 1. Each subsequent row is formed by adding adjacent numbers from the row above, with 1s at both ends. Row n contains the binomial coefficients C(n,0), C(n,1), …, C(n,n).
  • C(n,k) = n! / (k!(n-k)!) is the number of ways to choose k items from n. It appears at row n, position k (0-indexed) in Pascal's Triangle.
  • The sum of all entries in row n equals 2ⁿ. This is because each entry C(n,k) counts a subset, and the sum counts all subsets of an n-element set.
  • Σᵢ₌ᵣⁿ C(i,r) = C(n+1, r+1). Summing a diagonal of Pascal's Triangle gives the entry at the foot of the "hockey stick".
  • The sums of the shallow diagonals (going northeast) of Pascal's Triangle give the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, …

Related Calculators

Cₙ Catalan Numbers nCr Combinations & Permutations Fn Fibonacci Calculator 🎲 Probability Calculator
Sources & References (5) ▾
  1. An Introduction to the Theory of Numbers — Hardy & Wright — Oxford University Press
  2. Pascal's Arithmetical Triangle — A.W.F. Edwards — Johns Hopkins University Press
  3. Pascal's Triangle — Wolfram MathWorld — Wolfram Research
  4. Khan Academy — Pascal's Triangle and Binomial Expansion — Khan Academy
  5. Concrete Mathematics — Graham, Knuth & Patashnik (Chapter 5) — Addison-Wesley