Pascal's Rule

Theorem
For positive integers $$n, k \,\!$$ with $$1 \le k \le n \,\!$$:
 * $$\binom n {k-1} + \binom n k = \binom {n+1} k$$

where $$\binom n k$$ is a binomial coefficient.

This is also valid for the real number definition:


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

Direct Proof
Let $$n, k \in \N$$ with $$1 \leq k \leq n \,\!$$.

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Proof for Real Numbers
Follows directly from Factors of Binomial Coefficients:

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