Pascal's Triangle Calculator
Generate Pascal's triangle rows
How to Use This Calculator
Enter Number of Rows
Input the number of rows you want to generate (0 to 20). Row 0 contains just the number 1.
Generate Triangle
Click "Generate Triangle" to see Pascal's triangle displayed.
Explore Patterns
Notice the patterns: each number is the sum of the two numbers above it, and each row contains binomial coefficients.
Pascal's Triangle Properties
Each number = sum of two numbers directly above it
Row n contains binomial coefficients: C(n, k) for k = 0 to n
Construction Rules:
- First and last numbers in each row are always 1
- Each interior number is the sum of the two numbers above it
- Row n has (n+1) numbers
Example: Row 4
1, 4, 6, 4, 1
These are C(4,0), C(4,1), C(4,2), C(4,3), C(4,4)
Row 4 = 1 4 6 4 1
About Pascal's Triangle Calculator
Pascal's Triangle is a triangular array of numbers named after French mathematician Blaise Pascal. Each number is the sum of the two numbers directly above it. The triangle contains many interesting mathematical properties and patterns.
Key Properties
- Binomial Coefficients: Row n contains the coefficients for (a+b)ⁿ
- Symmetry: Each row is symmetric (reads the same left-to-right and right-to-left)
- Fibonacci Numbers: Sums of shallow diagonals give Fibonacci numbers
- Powers of 2: Sum of each row equals 2ⁿ
Applications
- Binomial expansion: (x+y)ⁿ coefficients
- Combinatorics: number of ways to choose items
- Probability: calculating combinations
- Number theory: exploring patterns and relationships
Frequently Asked Questions
What is Pascal's Triangle?
Pascal's Triangle is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it. It's named after Blaise Pascal, though it was known in other cultures earlier.
How do I read the triangle?
Start with row 0 containing just "1". Row 1 has "1 1". Each subsequent row adds another number, with each interior number being the sum of the two numbers above it.
What are binomial coefficients?
The numbers in row n of Pascal's Triangle are the binomial coefficients C(n,k), which tell you how many ways to choose k items from n items.
Why is there a row limit of 20?
Very large triangles become difficult to display and may cause performance issues. The limit of 20 rows provides a good balance for visualization and learning.
What's the connection to (a+b)ⁿ?
Row n gives the coefficients when expanding (a+b)ⁿ. For example, (a+b)³ = a³ + 3a²b + 3ab² + b³, and row 3 is 1, 3, 3, 1.