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Pascal's Formula

Each subsequent row of Pascal's triangle is obtained by adding the two entries diagonally above. This follows immediately from the binomial coefficient identity

(n; r)=(n!)/((n-r)!r!)
(1)
=((n-1)!n)/((n-r)!r!)
(2)
=((n-1)!(n-r))/((n-r)!r!)+((n-1)!r)/((n-r)!r!)
(3)
=((n-1)!)/((n-r-1)!r!)+((n-1)!)/((n-r)!(r-1)!)
(4)
=(n-1; r)+(n-1; r-1).
(5)

See also

Binomial Coefficient, Binomial Sums, Pascal's Triangle

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Cite this as:

Weisstein, Eric W. "Pascal's Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PascalsFormula.html

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