Math Tools

Pascal's Triangle Generator

Generate, explore, and discover patterns in Pascal's Triangle

Triangle Generator

Generate Pascal's Triangle up to a specified number of rows

Maximum 30 rows for display

Quick Generate

Binomial Coefficient Calculator

Calculate C(n,k) = n! / (k! × (n-k)!)

Binomial Coefficient Formula
C(n,k) = (nk) = n! / (k! × (n-k)!)
Calculate Multiple Values
Common Values

Row Finder

Get any specific row of Pascal's Triangle

Row 0 = [1], Row 1 = [1,1], Row 2 = [1,2,1], ...

Quick Lookup

Pattern Explorer

Discover hidden patterns in Pascal's Triangle

Famous Patterns in Pascal's Triangle
Natural Numbers

The second diagonal contains all natural numbers: 1, 2, 3, 4, 5, ...

Triangular Numbers

The third diagonal: 1, 3, 6, 10, 15, ... (sums 1+2+3+...)

Fibonacci Sequence

Sum shallow diagonals: 1, 1, 2, 3, 5, 8, 13, ...

Powers of 2

Row sums: 1, 2, 4, 8, 16, 32, ... (2ⁿ)

Powers of 11

First rows read as numbers: 1, 11, 121, 1331, 14641, ...

Sierpiński Triangle

Color odd numbers to reveal a fractal pattern

Pascal's Triangle Reference

Formulas, properties, and theoretical background

Recursive Definition
C(n,k) = C(n-1,k-1) + C(n-1,k)
Direct Formula
C(n,k) = n! / (k! × (n-k)!)
Symmetry Property
C(n,k) = C(n, n-k)
Key Properties
Row Sum

Sum of row n equals 2ⁿ

Σ C(n,k) = 2ⁿ

Binomial Theorem

(a+b)ⁿ = Σ C(n,k) × aⁿ⁻ᵏ × bᵏ

Hockey Stick

Sum along a diagonal equals the element below

Applications
Probability
  • Coin flip probabilities
  • Binomial distribution
  • Random walk analysis
Combinatorics
  • Counting combinations
  • Paths in a grid
  • Subset counting
Algebra
  • Polynomial expansion
  • Series coefficients
  • Generating functions
Historical Notes
Named After

Blaise Pascal (1623-1662), French mathematician

Also Known As

Yang Hui's Triangle (China, 13th century)
Tartaglia's Triangle (Italy)

First Description

Ancient India (Pingala, 2nd century BCE) - used in prosody

Understanding Pascal's Triangle

What Is Pascal's Triangle?

An infinite triangular array of binomial coefficients.

  • Each number is the sum of two above
  • Edges are always 1
  • Symmetrical left to right
  • Row n has n+1 elements
Reading the Triangle

Element at row n, position k equals C(n,k).

  • Row 0: [1]
  • Row 1: [1, 1]
  • Row 2: [1, 2, 1]
  • Row 3: [1, 3, 3, 1]
Practical Uses

Essential tool in mathematics and science.

  • Expanding (a+b)ⁿ
  • Probability calculations
  • Counting problems
  • Number theory proofs