Discrete Math help In TicTacToe we are given a 3 x 3 grid co

Discrete Math help.

In Tic-Tac-Toe, we are given a 3 x 3 grid, consisting of unmarked cells. Two players, Xavier and Olivia, take turns marking the cells of this grid. When it is Xavier?s turn, he chooses an unmarked cell and marks it with the letter X. Similarly, when it is Olivia?s turn, she chooses an unmarked cell and marks it with the letter O. The first turn is by Xavier. The players continue making turns until all cells have been marked. Below, you see an example of a completely marked grid. . What is the number of completely marked grids? Justify your answer. . What is the number of different ways (i.e., ordered sequences) in which the grid can be completely marked, when following the rules given above? Justify your answer.

Solution

a)

There are 2 possibilities for each grid, and there are 9 grids.

Hence, there are 2^9 = 512 possibilities. [ANSWER]

***********

b)

There has to be 4 O\'s and 5 X\'s, as X\'s go first.

Thus, by permutation of like objects, there are

P(9; 4,5) = 9!/4!5! = 126 ways. [ANSWER]