Binomial Coefficient/Examples/5 from -2

Example of Binomial Coefficient

$\dbinom {-2} 5 = -6$

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:

\(\ds \dbinom {-2} 5\)

\(=\)

\(\ds \dfrac { {-2}^{ \underline 5} } {5!}\)

\(\ds \)

\(=\)

\(\ds \dfrac {-2 \times -3 \times -4 \times -5 \times -6} {5 \times 4 \times 3 \times 2 \times 1}\)

Definition of Falling Factorial, Definition of Factorial

\(\ds \)

\(=\)

\(\ds -6\)

$\blacksquare$