ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Subset
N-subset
Set of N-subsets
Definition D2167
Binomial coefficient
Formulation 2
Let $n, m \in \mathbb{N}$ each be a D996: Natural number.
The binomial coefficient with respect to $(n, m)$ is the D996: Natural number \begin{equation} \binom{n}{m} : = |\mathcal{P}_m \{ 1, \ldots, n \}| \end{equation}
Formulation 3
Let $n, m \in \mathbb{N}$ each be a D996: Natural number.
The binomial coefficient with respect to $(n, m)$ is the D996: Natural number \begin{equation} \binom{n}{m} : = \# \left\{ E \subseteq \{ 1, \ldots, n \} : |E| = m \right\} \end{equation}
Children
D4197: Central binomial coefficient
Results
R3611
R5670
R2786: Complement property of binomial coefficient
R2787: Pascal's rule
R1831: Real arithmetic expression for binomial coefficient
R4945: Real binomial theorem for exponent five
R4944: Real binomial theorem for exponent four
R5172: Real binomial theorem for exponent seven
R5171: Real binomial theorem for exponent six
R4943: Real binomial theorem for exponent three
R3090: Scaling property of binomial coefficient
Conjectures
C15: Singmaster's conjecture