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If I'm not mistaken, N! | NN, as N! = 1*2*...*N, and NN = N*N*...*N. Therefore, NN/N! = NN-1/(N-1)!
The fact that NN and N! share a factor doesn't mean that N! is a divisor of NN, as the following table shows:
 N                 NN                 N!             NN % 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

In reply to Re^2: A bad shuffle by Anonymous Monk
in thread A bad shuffle by tlm

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