Pascal's Triangle Calculator

Calculate the value of a term in Pascal's triangle. The formula C(a, b) = a!/(b!(a - b)!), where a and b are non-negative integers, a ≥ b, and ! denotes the factorial, gives the value of row a, column b of Pascal's triangle. Rows and columns are zero-indexed: the top row is represented by a = 0 (and contains only a single value, C(0, 0) = 1) and the leftmost column, which is also entirely composed of 1s, is represented by b = 0. Values supported by this calculator are 0 ≤ b ≤ a ≤ 99999.

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Or calculate entire rows of Pascal's triangle (rows 0-99 allowed):

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Properties of Pascal's Triangle

• C(a, b), read "a choose b", represents the number of ways to choose an unordered subset of b elements from a set of a elements. For example, there are C(15, 5) = 3003 ways to select a starting lineup of 5 players from a basketball team of 15 players, not regarding the positions of the players.
• Pascal's triangle is symmetric along its central axis C(a, a/2). Therefore, C(a, b) = C(a, a - b).
• Each term in the triangle is the sum of the two terms directly above it: C(a, b) = C(a - 1, b) + C(a - 1, b - 1).
• C(a, b) = 1 if and only if b = 0 and/or a = b.
• Row n of Pascal's triangle gives the coeficients in the expansion of (x + y)ⁿ. For example, the 5th row of Pascal's triangle is {1, 5, 10, 10, 5, 1}, so (x + y)⁵ = x⁵ + 5x⁴y + 10x³y² + 10x²y³ + 5xy⁴ + y⁵.
• If each term is considered a digit, and terms greater than 9 are carried over, then row n in Pascal's triangle forms the value 11ⁿ. For example, row 4 of Pascal's triangle is {1, 4, 6, 4, 1}, and 11⁴ = 14641. Row 5 of the triangle is {1, 5, 10, 10, 5, 1}, which, when the numbers are treated as individual digits with the 10s carried over, forms 11⁵ = 161051.
• The sum of all numbers in row n of Pascal's triangle is equal to 2ⁿ.