Meaning of Generalized Binomial Coefficients

Question

The title speaks for itself. I have seen and understood that a common way to write binomial coefficients in general is $${n \choose k} = \frac{\Gamma{(n + 1)}}{\Gamma{(k + 1)}\Gamma{(n - k + 1)}},$$ or, to prevent singularities of the Gamma function, $${n \choose k} = \lim_{a \to n} \lim_{b \to k} \frac{\Gamma{(a + 1)}}{\Gamma{(b + 1)}\Gamma{(a - b + 1)}}.$$ However, what does this the value of this represent when either $n$ or $k$ is a non-integer? In other words, is there a combinatorial explanation for the binomial coefficient of real numbers in general?