How can one show $\displaystyle\frac{(m+n-1)!}{m ! (n-1)!}\leq \left[\frac{e (m+n-1)}{n}\right]^{n-1}$ ?



bound for binomial coefficients
Denote the quotient of the right and left hand sides,
$$
f(m,n)=\biggl(\frac{e(m+n-1)}n\biggr)^{n-1}\bigg/\binom{m+n-1}m.
$$
Then $f(m,1)=1$ for all $m\in\mathbb N$ and
$$
\frac{f(m,n+1)}{f(m,n)} 
=\frac{e}{\biggl(1+\dfrac1n\biggr)^n}\cdot\biggl(1+\frac1{m+n-1}\biggr)^{n-1} > 1,
$$
that is,
$$
f(m,n+1)> f(m,n)>\dots> f(m,2)> f(m,1)=1.
$$
This proves the required inequality. 


