Binomial Coefficient with Two/Corollary

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

• Closed Form for Triangular Numbers: Proof using Binomial Coefficients

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): $15$