Binomial Coefficient/Examples/2 from -5

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Example of Binomial Coefficient

$\dbinom {-5} 2 = 15$


Proof

From the definition of binomial coefficient:

$\dbinom r k = \begin {cases} \dfrac {r^{\underline k} } {k!} & : k \ge 0 \\ & \\ 0 & : k < 0 \end {cases}$

where $r^{\underline k}$ denotes the falling factorial.

Therefore:

\(\ds \dbinom {-5} 2\) \(=\) \(\ds \dfrac { {-5}^{ \underline 2} } {2!}\)
\(\ds \) \(=\) \(\ds \dfrac {-5 \times -6} {2 \times 1}\) Definition of Falling Factorial, Definition of Factorial
\(\ds \) \(=\) \(\ds 15\)

$\blacksquare$