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(B) $I. n! \leq n^n$ is true because $\frac{n^n}{n!} = \frac{n}{n} 	imes \frac{n}{n-1} 	imes \dots 	imes \frac{n}{1} \geq 1$.$II. (n!)^2 \leq n^n$

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$I, III$ and $IV$

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(B) $I. n! \leq n^n$ is true because $\frac{n^n}{n!} = \frac{n}{n} 	imes \frac{n}{n-1} 	imes \dots 	imes \frac{n}{1} \geq 1$.$II. (n!)^2 \leq n^n$ is false. For large $n$,$(n!)^2$ grows much faster than $n^n$.$III. 10^n \leq n!$ is true for $n > 1000$ because the product $\frac{n}{10} 	imes \frac{n-1}{10} 	imes \dots 	imes \frac{1}{10}$ will eventually exceed $1$ as $n$ increases.$IV. n^n \leq (2n)!$ is true. Since $(2n)! = 1 	imes 2 	imes \dots 	imes n 	imes (n+1) 	imes \dots 	imes 2n$,it is clearly much larger than $n^n = n 	imes n 	imes \dots 	imes n$ ($n$ times).Thus,$I, III,$ and $IV$ are true.

Among the inequalities below,which ones are true for all natural numbers $n > 1000$?
$I. n! \leq n^n$
$II. (n!)^2 \leq n^n$
$III. 10^n \leq n!$
$IV. n^n \leq (2n)!$

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Among the inequalities below,which ones are true for all natural numbers $n > 1000$?$I. n! \leq n^n$$II. (n!)^2 \leq n^n$$III. 10^n \leq n!$$IV. n^n \leq (2n)!$