How many Latin Squares?
Jelly, 8 bytes

Œ!ṗµfZ€L


Brute force solution. Inputs 5 and higher will time out on TIO.

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How it works

Œ!ṗµfZ€L  Main link. Argument: n

Œ!        Generate all permutations of R := [1, ..., n].
  ṗ       Take the n-th Cartesian power, yielding the array A of all n×n matrices
          whose rows are permutations of R.
   µ      New chain. Argument: A
     Z€   Zip each; transpose each matrix in A, yielding the array B of all n×n
          matrices whose columns are permutations of R.
    f     Filter; keep only matrices that appear in A and B.
       L  Compute the length, counting the kept matrices.