Pascal's Triangle
Pascal's Triangle reveals binomial coefficients and combinatorial patterns.
Learn the construction rules and formulas with examples.
The Formula
Pascal's Triangle is a triangular arrangement of numbers where each entry is the sum of the two entries directly above it. The triangle is named after French mathematician Blaise Pascal, who published a detailed study of it in 1653, though the pattern was known centuries earlier in China, India, and Persia.
Each row n of Pascal's Triangle contains the binomial coefficients C(n, 0), C(n, 1), ..., C(n, n). These coefficients appear in the expansion of (a + b)ⁿ, making the triangle a powerful tool for algebra and probability. The triangle also contains many hidden patterns: the Fibonacci sequence appears along its diagonals, and rows sum to powers of 2.
Row 0 is simply 1. Row 1 is 1, 1. Row 2 is 1, 2, 1. Row 3 is 1, 3, 3, 1. Row 4 is 1, 4, 6, 4, 1. Each edge is always 1, and every interior number is the sum of the two numbers above.
Variables
| Symbol | Meaning |
|---|---|
| C(n, k) | Binomial coefficient — the entry in row n, position k |
| n | Row number (starting from 0) |
| k | Position within the row (starting from 0) |
| n! | n factorial = n × (n-1) × (n-2) × ... × 1 |
Example 1
Find the value at row 6, position 2 of Pascal's Triangle.
Apply the formula: C(6, 2) = 6! / (2! × 4!)
6! = 720, 2! = 2, 4! = 24
C(6, 2) = 720 / (2 × 24) = 720 / 48
C(6, 2) = 15
Example 2
Use Pascal's Triangle to expand (x + y)⁴.
Row 4 of Pascal's Triangle: 1, 4, 6, 4, 1
Apply as coefficients: 1·x⁴y⁰ + 4·x³y¹ + 6·x²y² + 4·x¹y³ + 1·x⁰y⁴
(x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴
When to Use It
Pascal's Triangle is useful in many areas of mathematics.
- Expanding binomial expressions quickly
- Calculating combinations without a calculator
- Probability problems involving repeated trials
- Discovering number theory patterns and sequences