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Combinatorics Difficulty 3.5 AMC 10/12 Find the answer Canada

The letters A, B, C, D, and E are to be placed in the grid so that each of these letters appears exactly once in each row and exactly once in each column.

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

Which letter will go in the square marked with $*$ ?

Pick one

Solution

Each letter A, B, C, D, E appears exactly once in each column and each row.

The entry in the first column, second row cannot be A or E or B (the entries already present in that column) and cannot be C or A (the entries already present in that row).

Therefore, the entry in the first column, second row must be D.

This means that the entry in the first column, fourth row must be C.

The entry in the fifth column, second row cannot be D or C or A or E and so must be B.

This means that the entry in the second column, second row must be E.

Using similar arguments, the entries in the first row, third and fourth columns must be D and B, respectively.

This means that the entry in the second column, first row must be C.

Using similar arguments, the entries in the fifth row, second column must be A.

Also, the entry in the third row, second column must be D.

This means that the letter that goes in the square marked with $*$ must be B.

We can complete the grid as follows:

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

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.