# Pascal's Rule/Real Numbers

## Theorem

For positive integers $n, k$ with $1 \le k \le n$:

$\dbinom n {k - 1} + \dbinom n k = \dbinom {n + 1} k$

This is also valid for the real number definition:

$\forall r \in \R, k \in \Z: \dbinom r {k - 1} + \dbinom r k = \dbinom {r + 1} k$

## Proof

 $\displaystyle \left({r + 1}\right) \binom r {k - 1} + \left({r + 1}\right) \binom r k$ $=$ $\displaystyle \left({r + 1}\right) \binom r {k - 1} + \left({r + 1}\right) \binom r {r - k}$ Symmetry Rule for Binomial Coefficients $\displaystyle$ $=$ $\displaystyle k \binom {r + 1} k + \left({r - k + 1}\right) \binom {r + 1} {r - k + 1}$ Factors of Binomial Coefficient $\displaystyle$ $=$ $\displaystyle k \binom {r + 1} k + \left({r - k + 1}\right) \binom {r + 1} k$ Symmetry Rule for Binomial Coefficients $\displaystyle$ $=$ $\displaystyle \left({r + 1}\right) \binom {r + 1} k$

Dividing by $\left({r + 1}\right)$ yields the result.

$\blacksquare$

## Source of Name

This entry was named for Blaise Pascal.