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Trying to find a nice way to simplify the question:

Which is bigger 2000! or 1000^2000?

I don't know what kind of reasoning I can apply here.

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    $\begingroup$ Hint: compare 999*1001 to 1000^2 $\endgroup$
    – Lily
    Dec 3, 2014 at 0:25
  • $\begingroup$ Ah, that does it. I was trying to regroup in different ways but that definitely works, thanks. $\endgroup$
    – Bitmaximus
    Dec 3, 2014 at 0:28
  • $\begingroup$ Don't forget about 2000 :) $\endgroup$
    – Lily
    Dec 3, 2014 at 0:37

1 Answer 1

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In Factorial Inequality problem $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$ it is shown that

$$n!<\left(\frac n2\right)^n.$$ For $n=2000$ we have

$$2000!<\left(\frac {2000}{2}\right)^{2000}=1000^{2000}.$$

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