ThmDex – An index of mathematical definitions, results, and conjectures.
Result R2787 on D2167: Binomial coefficient
Pascal's rule
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}{m} = \binom{n - 1}{m} + \binom{n - 1}{m - 1} \end{equation}
Subresults
R5670: Pascal's rule in the upwards direction
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}
Applying R1831: Real arithmetic expression for binomial coefficient, we may directly compute \begin{equation} \begin{split} \binom{n - 1}{m} + \binom{n - 1}{m - 1} & = \frac{(n - 1)!}{(n - m - 1)! m!} + \frac{(n - 1)!}{(n - m)! (m - 1)!} \\ & = (n - 1)! \left( \frac{1}{(n - m - 1)! m!} + \frac{1}{(n - m)! (m - 1)!} \right) \\ & = (n - 1)! \left( \frac{1}{(n - m - 1)! m!} \frac{n - m}{n - m} + \frac{1}{(n - m)! (m - 1)!} \frac{m}{m} \right) \\ & = (n - 1)! \left( \frac{n - m}{(n - m)! m!} + \frac{m}{(n - m)! m!} \right) \\ & = (n - 1)! \frac{n}{(n - m)! m!} \\ & = \frac{n!}{(n - m)! m!} \\ & = \binom{n}{m} \\ \end{split} \end{equation} $\square$