*N*^{N-1}/(*N*-1)! is not an integer for any integer *N* > 2, so it is not the case that *N*! | *N*^{N}, except for *N*=1 and *N*=2.
the lowliest monk
**Update:** Here's a proof of the assertion made above. I'm sure there are better proofs of it, but this is the best I could come up with.
Assume that *N* > 2, and let *p* be the largest prime in the prime factorization of *N*. There are three cases to consider. Suppose first that *p* is 2. Then, by assumption, *N* is a power of 2 greater than or equal to 4. Therefore, 3 is a factor of *N*!, and consequently *N*! does not divide *N*^{N}. Next, suppose that *p* > 2. If *N* = *p*^{k} for some nonnegative integer *k*, then *N* is odd and not divisible by *N*! whose prime factorization includes 2. This leaves the case in which *p* > 2, and is not the sole prime factor of *N*. In this case *N* >= 2 *p*. By Bertrand's postulate there exists a prime *q* such that *p* < *q* < 2 *p* <= *N*. Therefore, there is a factor of *N*!, namely *q*, that does not divide *N*^{s}, for any positive integer *s*. Therefore, *N*! does not divide *N*^{N}, for all *N* > 2.
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If I'm not mistaken, N! | N^{N}, as N! = 1*2*...*N, and N^{N} = N*N*...*N. Therefore, N^{N}/N! = N^{N-1}/(N-1)!
The fact that N^{N} and N! share a factor doesn't mean that N! is a divisor of N^{N}, as the following table shows:
N N^{N} N! N^{N} % N!
1 1 1 0
2 4 2 0
3 27 6 3
4 256 24 16
5 3125 120 5
6 46656 720 576
7 823543 5040 2023
8 16777216 40320 4096
9 387420489 362880 227529
10 10000000000 3628800 2656000
11 285311670611 39916800 26301011
12 8916100448256 479001600 443667456
13 302875106592253 6227020800 5268921853
14 11112006825558016 87178291200 294332416
15 437893890380859375 1307674368000 820814907375
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My bad. Somehow the coffee fairy didn't stop by me, and I'm confusing "share a factor" with "evenly divides".
-QM
--
Quantum Mechanics: The dreams stuff is made of
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*How does this affect your comment?*
It doesn't. The important point is that `(N**N mod N!) != 0 for N > 2`, i.e., N**N balls can not be evenly distributed among N! buckets.
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