Question

True or false:

The binomial coefficient $\displaystyle{{-1/2}\choose{5}}$ is equal to $\dfrac{63}{256}$

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Answer 1

By definition, the binomial coefficient $\binom nk$ is given as follows:

$$\binom nk = \frac{n^{\underline{k}}}{k!} = \frac{n(n-1)(n-2)\cdots(n-(k-1))}{k(k-1)(k-2)\cdots 1}$$

Substituting $k = 5$ and $n = -1/2$, get:

$$\binom {-1/2}5 =\frac{(-1/2)^{\underline{5}}}{5!} =\\ = \frac{\left( -\dfrac12 \right)\left( -\dfrac12-1 \right)\left( -\dfrac12-2 \right)\left( -\dfrac12-3 \right)\left( -\dfrac12 -4\right)}{120} =\\ = -\dfrac{63}{256}$$

Answer. False.