Binomial Coefficient with Two/Corollary
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Theorem
- $\forall n \in \N: \dbinom n 2 = T_{n - 1} = \dfrac {n \paren {n - 1} } 2$
where $T_n$ is the $n$th triangular number.
Proof
From the definition of binomial coefficient:
- $\dbinom n 2 = \dfrac {n!} {2! \paren {n - 2}!}$
The result follows directly from the definition of the factorial:
- $2! = 1 \times 2$
$\blacksquare$
Also see
Sources
- 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.3$ Early Number Theory: Problems $1.3$: $2$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $15$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $15$