What Is Pascal's Triangle?
Pascal's Triangle is a triangular array of binomial coefficients named after French mathematician Blaise Pascal. Each number is the sum of the two numbers directly above it. The triangle starts with a single 1 at the top (row 0), and each subsequent row has one more element than the previous row.
Key Properties
Binomial Coefficients
Each entry is C(n, k), the number of ways to choose k items from n items.
C(n, k) = n! / (k!(n-k)!)
Row Sums
The sum of all elements in row n equals 2 raised to the power n.
Sum of row n = 2^n
Symmetry
Each row is symmetric. C(n, k) = C(n, n-k).
C(n, k) = C(n, n-k)
Fibonacci Numbers
Shallow diagonals of Pascal's Triangle sum to Fibonacci numbers.
1, 1, 2, 3, 5, 8, 13, ...
How to Build Pascal's Triangle
- Start with 1 at the top (row 0).
- Each row begins and ends with 1.
- Every other number is the sum of the two numbers above it from the previous row.
- Row n has (n + 1) elements.
Applications
- Combinatorics: Counting combinations and permutations.
- Binomial Expansion: Coefficients of (a + b)^n come from row n.
- Probability: Used in binomial probability distributions.
- Number Theory: Contains patterns related to prime numbers, triangular numbers, and more.
- Fractals: Coloring odd/even numbers produces the Sierpinski triangle pattern.