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Pascal's triangle

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Pascal's Triangle

Pascal's triangle is a triangular array where each entry is the sum of the two entries above it. Learn how to build it row by row, why row n gives the coefficients for expanding a binomial to the power n, and how each entry equals a combination C(n,k).

What Pascal's triangle is

Pascal's triangle is a triangular array of numbers where each row starts and ends with 1, and every number in between is the sum of the two numbers diagonally above it. It looks simple, but it encodes the coefficients used in the binomial theorem and counts combinations at the same time.

Pascal's triangle Six rows of Pascal's triangle: 1; 1,1; 1,2,1; 1,3,3,1; 1,4,6,4,1; 1,5,10,10,5,1. Each interior entry equals the sum of the two entries diagonally above it. row 0 row 1 row 2 row 3 row 4 row 5 11112113311464115101051
Rows 0 through 5 of Pascal's triangle. Each interior entry is the sum of the two entries above it.

Start row 0 with a single 1. To build the next row, place a 1 at each end, and fill each interior spot with the sum of the two numbers diagonally above it in the previous row. Row 3, for instance, comes from row 2 (1, 2, 1): the interior entries are 1+2=3 and 2+1=3, giving row 3 = 1, 3, 3, 1.

Row n of the triangle gives the coefficients for expanding a binomial raised to the power n. Row 3 (1, 3, 3, 1) is exactly the coefficients in (x + y)³ = x³ + 3x²y + 3xy² + y³. Each entry in row n is also the number of ways to choose k items from n — the same value as a combination, C(n, k).

You don't have to build every row from scratch. The k-th entry (starting from 0) in row n equals C(n, k) = n! ÷ (k!(n−k)!). For row 5, position 2 is C(5,2) = 10, which matches the triangle's row 5: 1, 5, 10, 10, 5, 1.

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