Is the binomial coefficient linear?

Question

Consider the binomial coefficient

B = $\frac {(c + n -1)!}{c!(n-1)!}$

If we increase the value of c, will we always get a higher value B? Likewise, if we increase the value of n, will we always get a higher value B?

I want to graph this, but i don't know how to accomplish that. So my question is, is the coefficient linear depending on the values of c and n?

Answer 1

Increasing $c$ or $n$ will increase $B$, as you are going down Pascal's triangle by doing so. However, the one-variable function obtained by fixing $c$ or $n$ is not linear unless $c=1$ or $n-1=1$.

$\binom nk$ for fixed $k$ is a polynomial of degree $k$.

Answer 2

Consider $$\log(B)=\log \left(\frac{(c+n-1)!}{c! (n-1)!}\right)=\log \left(\frac{\Gamma (c+n)}{\Gamma (c+1) \Gamma (n)}\right)$$ and admit that $c$ is a continous variable, you havr $$\frac {B'} B=H_{c+n-1}-H_c >0$$ So, $B$ increases with $c$.