ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5670 on D2167: Binomial coefficient
Subresult of R2787: Pascal's rule
Pascal's rule in the upwards direction
Formulation 0
Let $n, m \in \mathbb{N}$ each be a D996: Natural number such that
(i) \begin{equation} 1 \leq m \leq n \end{equation}
Then \begin{equation} \binom{n + 1}{m} = \binom{n}{m} + \binom{n}{m - 1} \end{equation}
Proofs
Proof 0
Let $n, m \in \mathbb{N}$ each be a D996: Natural number such that
(i) \begin{equation} 1 \leq m \leq n \end{equation}
This result is a direct corollary to R2787: Pascal's rule. $\square$