2021 Fall AMC 12A Problem 22
Problem 22 of 25HarderCounting & Probability
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?
Answer choices
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Solution
Carl wins on his third so the board has three s forming one of the lines and three s in the other six cells. The s must not form a line (else Azar would have won first).
If the line is a row or column ( choices), the remaining six cells contain two full lines, so valid placements number If the line is a diagonal ( choices), the remaining six cells contain no full line, giving
The total is
Thus, the correct answer is D.