AMC 8, 2022, Problem Marathon

[Arisha Roy]

Solution

First, we will only consider cases where the three identical symbols are in the same column.

Then we will double our answer as the same holds true for rows.

There are 3 ways to choose a column with all \(\bigcirc\) 's and 2 ways to choose a column with all \(\triangle\) 's.

The third column can be filled in \(2^3=8\) ways.

Therefore, we have a total of \(3 \times 2 \times8=48\) cases.

But here we overcounted the cases with 2 columns of  one symbol and 1 column of another symbol.

This happens in \(2 \times 3=6\) cases.

Then our answer considering the columns will be \(48-6=42\).

Total number of configuration will be \(42 \times 2=84\)

• This reply was modified 1 year, 5 months ago by Arisha Roy.

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