# Prove that $n! ≥ (⌈n/2⌉)^{⌈n/2⌉}$

Prove that : $n! ≥ (⌈n/2⌉)^{⌈n/2⌉}$

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Hint: Count it. How many integers satisfies $k\ge n/2$? – tetori Feb 3 at 8:57
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$$n! \geq n(n-1)(n-2)(n-3) \cdots \lceil n/2 \rceil \geq (\lceil n/2 \rceil)^{\lceil n/2 \rceil}$$
Hint: $2a*(2a-1)*(2a-2)*...*a*k > a^a$ where $k>0$.