Pascal's Rule/Real Numbers

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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.


Sources