Four players are playing a game involving choosing squares on a grid of size 3×8. Each player chooses a random square on the grid, then all players reveal their choices and a token is placed in the center of each of these squares.
The probability that the tokens form the vertices of a "non-degenerate" rectangle can be expressed as a/b where a and b are coprime positive integers. What is the value of a+b?
Note:
1. Players are allowed to have selected the same squares.
2. There is no restriction on their choices.
3. A degenerate rectangle has 0 area.
4. Squares are rectangles.
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To find the probability that the tokens form the vertices of a non-degenerate rectangle, let's consider the different possibilities for forming a rectangle.
A non-degenerate rectangle will have unequal side lengths (width and height). In this case, there are two options for side lengths:
1. Width = 1 and height = 2.
2. Width = 2 and height = 1.
Let's calculate the probability for each case:
Case 1: Width = 1 and height = 2
For this case, we need two squares to be chosen vertically adjacent to each other and two other squares to be chosen horizontally adjacent to each other.
The probability of choosing two vertically adjacent squares is (3 × 8) / (3 × 8) = 1. Since all players can choose the same square, there is no restriction on their choices.
Similarly, the probability of choosing two horizontally adjacent squares is also 1.
Therefore, the probability for this case is 1 * 1 = 1.
Case 2: Width = 2 and height = 1
For this case, we need two squares to be chosen horizontally adjacent to each other and two other squares to be chosen vertically adjacent to each other.
The probability of choosing two horizontally adjacent squares is the same as in Case 1, i.e., 1.
Similarly, the probability of choosing two vertically adjacent squares is also 1.
Therefore, the probability for this case is also 1 * 1 = 1.
Since these two cases are mutually exclusive, we can add their probabilities together: 1 + 1 = 2.
Now, we have the probability as 2/1, and since 2 and 1 are already coprime, a = 2 and b = 1.
To find the value of a + b, we just need to add them: 2 + 1 = 3.
Therefore, the value of a + b is 3.
Note: The fact that squares can be chosen more than once allows for the possibility of forming rectangles. If no squares could be chosen multiple times, the probability would be 0.