2010 AIME II Problems/Problem 11
Problem 11
Define a T-grid to be a matrix which satisfies the following two properties:
- Exactly five of the entries are 's, and the remaining four entries are 's.
- 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.
- 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.
- Both takes a column: Using the simlar reasoning, there are cases.
Case : cases
Case : Player wins twice.
- A row and a column
- ways to pick the row,
- to pick the column.
There are cases
- A row/column and a diagonal
- ways to pick the row/column,
- to pick the diagonal.
There are cases
- 2 diagonals It is clear that there is only case.
Case total:
Thus, the answer is
See also
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 | ||
All AIME Problems and Solutions |