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 | |||||||||||
\(\ds \) | \(=\) | \(\ds 15\) |
$\blacksquare$