Orthogonal Latin Squares of Order 5/Mistake

Source Work

 * The Puzzles:
 * The Thirty-six Officers Problem
 * $135$
 * $135$

Mistake

 * How can five each of As, Bs, Cs, Ds and Es be placed in these cells so that no letter is repeated in any row or column?

$\begin{array} {|c|c|c|c|c|} \hline \ \ & \ \ & \ \ & \ \ & \ \ \\ \hline &  &  &  &  \\ \hline &  &  &  &  \\ \hline &  &  &  &  \\ \hline &  &  &  &  \\ \hline \end{array}$

Correction
There is more to it than just arranging the letters so that no letter is repeated in any row or column.

We need to place $2$ letters in each space so that no letter of either pair is repeated in any row or column.

Otherwise we just get something like this boring arrangement:

$\begin{array} {|c|c|c|c|c|} \hline A & B & C & D & E \\ \hline B & C & D & E & A \\ \hline C & D & E & A & B \\ \hline D & E & A & B & C \\ \hline E & A & B & C & D \\ \hline \end{array}$

which is not what is being asked.