My PhD supervisor called the following result something along the lines: "The theorem in the theory of Latin squares with the most incorrect proofs" (then, to my embarrassment, proceeded to point out an error in my "proof" at the time). I'll pose proving it as a problem.
Definition: A $k \times n$ *Latin rectangle* is a $k \times n$ matrix with symbols from $\{1,2,\ldots,n\}$ for which each symbol occurs exactly once in each row and at most once in each column. For example:
1 2 4 3
2 3 1 4
4 1 3 2
is a $3 \times 4$ Latin rectangle.
Definition: A Latin rectangle is called reduced if the first row is $(1,2,\ldots,n)$ and the first column is $(1,2,\ldots,k)^T$.
The above Latin rectangle is not reduced, but this one is:
1 2 3 4
2 3 4 1
3 4 1 2
Let $L_{k,n}$ be the number of $k \times n$ Latin rectangles. Let $R_{k,n}$ be the number of reduced $k \times n$ Latin rectangles.
Theorem: For all $1 \leq k \leq n$, \[L_{k,n}=\frac{n!(n-1)!}{(n-k)!}R_{k,n}.\]
Problem: Prove it (and double-check your proof).