Pascal's rule

1. Sablon:Matematika ?

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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, $$(\genfrac{}{}{0ex}{}{n-1}{k})+(\genfrac{}{}{0ex}{}{n-1}{k-1})=(\genfrac{}{}{0ex}{}{n}{k}),$$ where $(\genfrac{}{}{0ex}{}{n}{k})$ is a binomial coefficient; one interpretation of the coefficient of the $1|$ term in the expansion of $1|$. There is no restriction on the relative sizes of Sablon:Mvar and Sablon:Mvar, since, if $1|$ 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 $$\frac{(x+y)!}{x!y!}=(\genfrac{}{}{0ex}{}{x+y}{x})=(\genfrac{}{}{0ex}{}{x+y}{y})$$ solves the linear two-dimensional difference equation $$N_{x,y}=N_{x-1,y}+N_{x,y-1},N_{0,y}=N_{x,0}=1$$ 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.