Evaluating a Negative Fractional Binomial Coefficient Using Factorial Notation
Using factorial notation, evaluate the binomial coefficient \(\binom{-\frac{1}{2}}{3}\) and simplify your answer. Recall that the binomial coefficient for non-integer upper indices is defined by \(\binom{k}{n} = \frac{k(k-1)(k-2)\cdots (k-n+1)}{n!}\) for an integer \(n \geq 1\).