Pascal's rule
Combinatorial identity about binomial coefficients / From Wikipedia, the free encyclopedia
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In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients. It states that for positive natural numbers n and k,
where is a binomial coefficient; one interpretation of the coefficient of the xk term in the expansion of (1 + x)n. There is no restriction on the relative sizes of n and k,[1] since, if n < k the value of the binomial coefficient is zero and the identity remains valid.
Pascal's rule can also be viewed as a statement that the formula
solves the linear two-dimensional difference equation
over the natural numbers. Thus, Pascal's rule is also a statement about a formula for the numbers appearing in Pascal's triangle.
Pascal's rule can also be generalized to apply to multinomial coefficients.