Pascal's rule

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,

{n-1 \choose k}+{n-1 \choose k-1}={n \choose k},

where ${\tbinom {n}{k}}$ is a binomial coefficient; one interpretation of the coefficient of the $x k$ term in the expansion of $(1 + x) n$ . There is no restriction on the relative sizes of $n$ and $k$ , 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

{\frac {(x+y)!}{x!y!}}={x+y \choose x}={x+y \choose y}

solves the linear two-dimensional difference equation

N_{x,y}=N_{x-1,y}+N_{x,y-1},\quad 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.

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