2010 AIME II Problems/Problem 11

Problem

Define a T-grid to be a matrix which satisfies the following two properties:

1. Exactly five of the entries are 's, and the remaining four entries are 's.
2. Among the eight rows, columns, and long diagonals (the long diagonals are and , 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 's and four 's.

There are ways to fill the board with five 's and four 's. Now we need to subtract the number of bad grids.

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

Case : 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.

1. Both takes a row:
   • ways for player to pick a row,
   • ways for player ,
   • ways for player to take a single box in the remaining row.

   There are cases.

2. Both takes a column: Using the simlar reasoning, there are cases.

Case : cases

Case : Player wins twice.

1. A row and a column
   • ways to pick the row,
   • to pick the column.

   There are cases

2. A row/column and a diagonal
   • ways to pick the row/column,
   • to pick the diagonal.

   There are cases

3. 2 diagonals It is clear that there is only case.

Case total:

Thus, the answer is