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My approach:: Here $A>C$ and Using the formula $\displaystyle \left(\frac{n}{2}\right)^n>n! >\left(\frac{n}{3}\right)^n$ for all natural no. $n>6$

and put $n=600,$ we get $(300)^{600}>600!>(200)^{600}$ So $A>B>C$

My Question is How can we prove $\displaystyle \left(\frac{n}{2}\right)^n>n! >\left(\frac{n}{3}\right)^n$ for all natural no. $n>6$

Can we prove it without induction.

Thanks