Difference between revisions of "2021 Fall AMC 12A Problems/Problem 22"
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==Problem== | ==Problem== | ||
| − | Azar and Carl play a game of tic-tac-toe. Azar places an in <math>X</math> one of the boxes in a 3-by-3 array of boxes, then Carl places an <math>O</math> in one of the remaining boxes. After that, Azar places an <math>X</math> in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third <math>O</math>. How many ways can the board look after the game is over? | + | Azar and Carl play a game of tic-tac-toe. Azar places an in <math>X</math> one of the boxes in a <math>3</math>-by-<math>3</math> array of boxes, then Carl places an <math>O</math> in one of the remaining boxes. After that, Azar places an <math>X</math> in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third <math>O</math>. How many ways can the board look after the game is over? |
<math>\textbf{(A) } 36 \qquad\textbf{(B) } 112 \qquad\textbf{(C) } 120 \qquad\textbf{(D) } 148 \qquad\textbf{(E) } 160</math> | <math>\textbf{(A) } 36 \qquad\textbf{(B) } 112 \qquad\textbf{(C) } 120 \qquad\textbf{(D) } 148 \qquad\textbf{(E) } 160</math> | ||
Revision as of 16:25, 28 November 2021
Problem
Azar and Carl play a game of tic-tac-toe. Azar places an in 
one of the boxes in a 
-by- array of boxes, then Carl places an 
in one of the remaining boxes. After that, Azar places an in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third . How many ways can the board look after the game is over?
Solution
We need to find out the number of configurations with 3 and 3 with 3 in a row, and 3 not in a row.
: 3 are in a horizontal row or a vertical row.
Step 1: We determine the row that 3 occupy.
The number of ways is 6.
Step 2: We determine the configuration of 3 .
The number of ways is 
.
In this case, following from the rule of product, the number of ways is 
.
: 3 are in a diagonal row.
Step 1: We determine the row that 3 occupy.
The number of ways is 2.
Step 2: We determine the configuration of 3 .
The number of ways is 
.
In this case, following from the rule of product, the number of ways is 
.
Putting all cases together, the total number of ways is 
.
Therefore, the answer is 
.
~Steven Chen (www.professorchenedu.com)
| 2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 21 |
Followed by Problem 23 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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