> [!NOTE] > For all positive integers $n,$ $n! \geq \left( \frac{n}{2} \right)^{n/2}$where $n!$ denotes the [[Factorial|factorial]]. ###### Proof Suppose $n$ is even, say $n=2m.$ Then $n!=\underbrace{(2m)(2m-1)\ldots(m+1)}_{m\mathrm{~factors}}m!\geq(2m)(2m-1)\ldots(m+1)>m^m=\left(\frac n2\right)^{n/2}.$ Now suppose $n=2m+1.$ Then $n!=\underbrace{(2m+1)(2m)\ldots(m+1)}_{m+1\mathrm{~factors}}m!\geq(m+1)^{m+1}>\left(\frac n2\right)^{n/2}.$