I have following task:
Let $d,m \in \mathbb{N} $ and we define $\tau \geq e^{\pi - 1}$. Then it follows $ \binom{m + d}{m} \geq (\frac{m}{\tau d})^d$.
In a paper the inequality is mentioned without any further explaination except we use Stirling's formula.
Neither I haven't found any formula to explain this nor I can easily change Stirling's formula to fit.
Do you have any proposal?
As $\tau\ge e^{\pi-1}\approx 8.5\gg 1$, $${m+d\choose m}={m+d\choose d}=\frac{(m+d)(m+d-1)\cdots (m+1)}{d(d-1)\cdots 1}>\frac{m\cdot m\cdots m}{d\cdot d\cdots d} =\left(\frac md\right)^d>\left(\frac m{\tau d}\right)^d$$