Problem:
A or is placed in each of the nine squares in a grid. Shown below is a sample configuration with three 's in a line.
How many configurations will have three 's in a line and three 's in a line? To determine the number of configurations with three 's in a line and three 's in a line?
Answer Choices:
A.
B.
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E.
Solution:
The and lines must have the same orientation: they are either in rows or in columns. First assume that the lines are in rows; the case for columns gives the same number of configurations. There are two cases to consider: either there is exactly one row of each type, or there are two rows of one type and one row of the other.
There is exactly one row and one row. There are ways to select the positions of the two rows. For the remaining row, there are total arrangements, however the symbols cannot be all s or all s, leaving possible arrangements. The number of configurations is therefore .
There are two rows of one type and one row of the other. There are ways to choose the positions of the identical rows and ways to choose the symbol for them. The third row is then determined. The number of configurations is .
In total there are configurations with and in rows. Doubling to account for the column orientation gives a total of configurations that have three s in a line and three s in a line.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions