# Prove that $n! ≥ (⌈n/2⌉)^{⌈n/2⌉}$ [closed]

Prove that : $n! ≥ (⌈n/2⌉)^{⌈n/2⌉}$

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## closed as off-topic by This is much healthier., Daniel Rust, Hakim, glace, hardmathJul 4 at 0:47

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Hint: Count it. How many integers satisfies $k\ge n/2$? –  tetori Feb 3 '13 at 8:57
Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. If this is homework, please add the homework tag; people will still help, so don't worry. Also, many find the use of imperative ("Prove", "Solve", etc.) to be rude when asking for help; please consider rewriting your post. –  Julian Kuelshammer Feb 3 '13 at 9:19

$$n! \geq n(n-1)(n-2)(n-3) \cdots \lceil n/2 \rceil \geq (\lceil n/2 \rceil)^{\lceil n/2 \rceil}$$

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Hint: $2a*(2a-1)*(2a-2)*...*a*k > a^a$ where $k>0$.

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