The Pascal's Triangle Calculator generates Pascal's triangle up to any number of rows (up to 20 for performance). Pascal's triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. It is named after the French mathematician Blaise Pascal, although it was known to ancient Chinese and Persian mathematicians centuries earlier. This triangle encodes binomial coefficients, combinations C(n,k), and plays a crucial role in algebra, probability theory, and combinatorics.
How Pascal's Triangle is Constructed
Step 1: Start with a single 1 at the top (row 0).
Step 2: Each row begins and ends with 1.
Step 3: Each interior number is the sum of the two numbers directly above it (from the previous row).
Step 4: Continue to build subsequent rows.
Mathematical Properties & Patterns
- Binomial coefficients: Row n contains C(n,0), C(n,1), ..., C(n,n).
- Row sums: The sum of numbers in row n is 2ⁿ.
- Fibonacci sequence: Sum of shallow diagonals gives Fibonacci numbers.
- Hockey‑stick pattern: The sum of numbers along a diagonal equals the number below the last one.
- Powers of 11: Row n interpreted as a single number gives 11ⁿ (with carrying).
Applications
- Binomial expansion: Coefficients of (a+b)ⁿ come directly from row n.
- Probability (combinations): C(n,k) is the number of ways to choose k items from n.
- Algebra: Expanding powers of binomials.
- Computer science: Dynamic programming, probability trees.