Pascal's Triangle Calculator
Generate Pascal's Triangle and explore binomial coefficients.
Table of Contents
How to Use
- Enter the number of rows you want to generate (1-20).
- Click Calculate to generate Pascal's Triangle.
- View the triangle, row sums, and binomial expansion coefficients.
- Use the coefficients for binomial expansions like (a + b)^n.
What Is Pascal's Triangle?
Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The triangle starts with a single 1 at the top, and each subsequent row begins and ends with 1.
Named after French mathematician Blaise Pascal, this arrangement reveals fascinating patterns in combinatorics, probability theory, and algebra.
Key Properties
- Each row sum equals 2^n where n is the row number (starting from 0)
- The entries in row n are the binomial coefficients C(n,k)
- The triangle is symmetric - each row reads the same forwards and backwards
- Diagonal patterns reveal Fibonacci numbers, triangular numbers, and more
Applications
| Field | Application |
|---|---|
| Algebra | Binomial expansion coefficients for (a + b)^n |
| Probability | Calculating combinations and probabilities |
| Combinatorics | Counting paths and arrangements |
| Number Theory | Exploring divisibility patterns |
Frequently Asked Questions
- How do I read Pascal's Triangle?
- Start at the top with 1. Each number below is the sum of the two numbers above it. The edges are always 1. Row numbers start at 0, so row 0 has one 1, row 1 has two 1s, and so on.
- What are binomial coefficients?
- Binomial coefficients are the numbers in Pascal's Triangle. They represent the coefficients when expanding (a + b)^n. For example, row 3 (1, 3, 3, 1) gives the coefficients for (a + b)³ = a³ + 3a²b + 3ab² + b³.
- Why do row sums equal powers of 2?
- Each row sum equals 2^n because it represents all possible combinations of choosing items from n objects. This is equivalent to (1 + 1)^n = 2^n when you substitute a = b = 1 in the binomial expansion.
- What patterns exist in Pascal's Triangle?
- Many patterns exist: the Fibonacci sequence appears along diagonals, triangular numbers appear in the third diagonal, and the sum of each row is a power of 2. The triangle also shows symmetry and fractal patterns when colored by divisibility.