2010 AIME II Problems/Problem 11

Problem

Define a T-grid to be a $3\times3$ matrix which satisfies the following two properties:

Exactly five of the entries are $1$ 's, and the remaining four entries are $0$ 's.
Among the eight rows, columns, and long diagonals (the long diagonals are $\{a_{13},a_{22},a_{31}\}$ and $\{a_{11},a_{22},a_{33}\}$ , no more than one of the eight has all three entries equal.

Find the number of distinct T-grids.

Solution

The T-grid can be consider as a tic-tac-toe board: five $1$ 's and four $0$ 's.

There are $\dbinom{9}{5} = 126$ ways to fill the board with five $1$ 's and four $0$ 's. Now we need to subtract the number of bad grids.

Let three-in-a-row/column/diagonal be a "win" and let player $0$ be the one that fills in $0$ and player $1$ fills in $1$ .

Case $1$ : Each player wins once.

If player takes a diagonal, the other cannot win, and if either takes a row/column, all column/row are blocked, so they either both take a row or both take a column.

Both takes a row:
- $3$ ways for player $1$ to pick a row,
- $2$ ways for player $0$ ,
- $3$ ways for player $0$ to take a single box in the remaining row.
There are $18$ cases.
Both takes a column: Using similar reasoning, there are $18$ cases.

Case $1$ : $36$ cases

Case $2$ : Player $1$ wins twice.

A row and a column
- $3$ ways to pick the row,
- $3$ to pick the column.
A row/column and a diagonal
- $6$ ways to pick the row/column,
- $2$ to pick the diagonal.
2 diagonals It is clear that there is only $1$ case.

Case $2$ total: $22$

Thus, the answer is $126-22-36=\boxed{068}$

2010 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2010 AIME II Problems/Problem 11

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