Pascal's Triangle Calculator
Generate rows of Pascal's triangle or look up any individual binomial coefficient C(n, k). Useful for combinatorics, probability problems, and expanding binomial expressions quickly.
About this calculator
Pascal's triangle is a triangular array where each entry is the sum of the two entries directly above it. Every entry equals the binomial coefficient C(n, k) = n! / (k! × (n−k)!), where n is the row index (starting at 0) and k is the position within that row (also starting at 0). For example, row 4 reads 1, 4, 6, 4, 1 — matching C(4,0) through C(4,4). The triangle appears throughout mathematics: the coefficients of (a + b)ⁿ are exactly row n, and each row's entries sum to 2ⁿ. Rather than computing full factorials, the calculator uses the multiplicative formula: start with 1, then for each step i from 0 to k−1 multiply by (n − i) and divide by (i + 1), rounding to the nearest integer to avoid floating-point drift.
How to use
Suppose you want the binomial coefficient at row 6, position 2 — that is C(6, 2). Set Output Type to 'coefficient', enter Specific Row (n) = 6 and Specific Position (k) = 2. The calculator computes: start = 1; step 0 → 1 × (6−0)/(0+1) = 6; step 1 → 6 × (6−1)/(1+1) = 15. Result: 15. To verify with the full triangle, switch Output Type to 'triangle', set Number of Rows to 7, and confirm row 6 reads 1, 6, 15, 20, 15, 6, 1 — the third entry (position 2) is indeed 15.
Frequently asked questions
How do you calculate a specific binomial coefficient using Pascal's triangle?
Every entry in Pascal's triangle is a binomial coefficient C(n, k), where n is the row number and k is the position within that row, both counted from zero. You can compute it with the formula C(n, k) = n! / (k! × (n−k)!), or more efficiently with the multiplicative method: multiply n × (n−1) × … × (n−k+1) and divide by k!. For example, C(5, 3) = (5 × 4 × 3) / (3 × 2 × 1) = 10. This value appears at row 5, position 3 of the triangle.
What are the practical uses of Pascal's triangle in probability and algebra?
In algebra, the entries of row n give the coefficients when expanding (a + b)ⁿ, saving you from tedious multiplication. In probability, C(n, k) counts the number of ways to choose k successes from n trials, which is the core of the binomial distribution formula. Pascal's triangle also reveals patterns like triangular numbers (column 2), powers of 2 (row sums), and the Fibonacci sequence hidden in its diagonals. These properties make it a fundamental reference tool in discrete mathematics and statistics courses.
Why does Pascal's triangle start at row 0 and position 0 instead of row 1 and position 1?
The zero-based indexing aligns the triangle directly with the binomial coefficient notation C(n, k): the very tip of the triangle is C(0, 0) = 1, which represents the expansion (a + b)⁰ = 1. Starting at 1 would introduce an offset that makes formulas more complicated and error-prone. Zero-based indexing also matches how most programming languages implement arrays, making the triangle easy to generate with loops. Mathematically, it is the most natural convention because it keeps the identity C(n, 0) = 1 consistent across all rows without special cases.