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:

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

$\ds \paren {r + 1} \binom r {k - 1} + \paren {r + 1} \binom r k$

$=$

$\ds \paren {r + 1} \binom r {k - 1} + \paren {r + 1} \binom r {r - k}$

$\ds $

$=$

$\ds k \binom {r + 1} k + \paren {r - k + 1} \binom {r + 1} {r - k + 1}$

$\ds $

$=$

$\ds k \binom {r + 1} k + \paren {r - k + 1} \binom {r + 1} k$

$\ds $

$=$

$\ds \paren {r + 1} \binom {r + 1} k$

Dividing by $\paren {r + 1}$ yields the result.

$\blacksquare$

This entry was named for Blaise Pascal.

1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $\text{D} \ (9)$