Best fit curve associated with the combination formula

[iScience]

Sorry if this is more of a HW question (if so then moderator please move my question. Thanks!)

Hi, I'm trying to get an expression for a best fit curve of the combination formula ($_nC_r$).
As far as I can tell, the curve is a simple parabolic curve, and its shape doesn't change. It's just scaled depending on $n$.

How might I get the continuous form of the discrete combination formula?

 

[mfb]

It is not a parabola.

The generalization of factorials to the real numbers is the gamma function.