Simplify symbolic inequality?

Questions
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1

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“The number of k-permutations is always less (or equal) than the numer of k-tupels”, is a known simple truth in combinatorics:

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Table[n!/(n - k)! <= n^k, {n, 1, 100}, {k, 1, n}] (* Out= {True,True,True,etc} *)

I’ve tried combinations of Simplify, FullSimplify, Reduce, FunctionExpand, Assuming, Assumptions and can’t get a True (i actually once got a False). A failing example with Reduce is:

Reduce[n!/(n - k)! <= n^k && n > k && k > 0, {n, k}, Integers]

Does anyone have better luck in getting a True for this inequality expression?

2

Using: Mathematica 12.2.0

Maybe:

 AsymptoticLessEqual[n!/(n - k)!, n^k, n -> \[Infinity], 
 Assumptions -> {k \[Element] Integers, k > 0}]
 (*True*)

But:

  AsymptoticLessEqual[n!/(n - k)!, n^k, k -> \[Infinity], 
  Assumptions -> {n \[Element] Integers, n > 0}]
  (*Can't find ?*)

or:

AsymptoticLessEqual[n!/(n - k)!, n^k, {n, k} -> \[Infinity]]
(*Can't....*)
3

Interesting approach, thank you! Your True is only valid for Infinity, not for normal values, as shown in the graph. Looks like the expression is a challenge for the Wolfram engine.