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Back to Combinatorics

Pascal's Triangle

Built with nothing but addition, yet every entry is a binomial coefficient C(n, k). Its symmetry, its diagonals and its row sums are all counting facts in disguise.

Step 1 of 18

Speed

Row 0 is a single 1. Every row after it starts and ends with 1, and every number in between is the sum of the two numbers just above it.

\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}, \qquad \binom{n}{0} = \binom{n}{n} = 1

Rows shown1n (row)0k (position)0

Rows and positions are counted from 0. Clicking a cell sets n and k; moving the sliders moves the highlight.

Pascal's Triangle

Built with nothing but addition, yet every entry is a binomial coefficient C(n, k). Its symmetry, its diagonals and its row sums are all counting facts in disguise.

Step 1 of 18

Speed

Row 0 is a single 1. Every row after it starts and ends with 1, and every number in between is the sum of the two numbers just above it.

\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}, \qquad \binom{n}{0} = \binom{n}{n} = 1

Rows shown1n (row)0k (position)0

Rows and positions are counted from 0. Clicking a cell sets n and k; moving the sliders moves the highlight.

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Pascal's Triangle

See also

Pascal's Triangle

See also