Pascal's Triangle Calculator
Generate and Explore Pascal's Triangle
Generate Pascal's Triangle with beautiful visualizations and explore its mathematical properties.
Generate Triangle
Enter between 1 and 30 rows
Construction Rule:
Each number = sum of two numbers above it
C(n,k) = C(n-1,k-1) + C(n-1,k)
What is Pascal's Triangle?
Pascal's Triangle is a triangular array of binomial coefficients. Each number is the sum of the two directly above it.
Construction:
Start with 1 at the top
Each row starts and ends with 1
Interior numbers = sum of two above
Formula:
C(n,k) = n! / (k! × (n-k)!)
Features
- Generate up to 30 rows
- Interactive cell selection
- Color-coded patterns
- Row sum calculations
- Binomial coefficients
- Copy to clipboard
- Download as text file
- Mathematical properties
Properties
Symmetry:
Triangle is symmetric about vertical axis
Row Sum:
Sum of row n = 2ⁿ
Diagonals:
First diagonal: all 1s
Second diagonal: natural numbers
Third diagonal: triangular numbers
Fibonacci:
Diagonal sums form Fibonacci sequence
Applications
Combinatorics:
Binomial coefficients, combinations
Algebra:
Binomial theorem expansion
Probability:
Binomial distribution
Number Theory:
Patterns in modular arithmetic
Patterns in Pascal's Triangle
Number Patterns:
- Natural numbers (1, 2, 3, 4...) in 2nd diagonal
- Triangular numbers (1, 3, 6, 10...) in 3rd diagonal
- Tetrahedral numbers in 4th diagonal
- Powers of 2 in row sums
- Powers of 11 in first few rows (base 10)
Use Cases
- Binomial expansion coefficients
- Probability calculations
- Combinatorics problems
- Polynomial expansion
- Number theory research
- Mathematical education
- Algorithm analysis
- Fractal patterns (Sierpinski triangle)
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