Four players are playing a game involving choosing positions on a grid of size 3×8. Each player chooses a random position on the grid, then all players reveal their choices and a token is placed on each of the positions. 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?
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To find the probability that the tokens form the vertices of a non-degenerate rectangle, we need to determine the total number of possible ways the players can choose their positions, as well as the number of ways that result in forming a non-degenerate rectangle.
First, let's calculate the total number of ways the players can choose their positions. Since there are 3 rows and 8 columns on the grid, each player has a total of 3 × 8 = 24 positions to choose from. Therefore, the total number of ways the players can choose their positions is 24^4, since there are four players.
Next, let's determine the number of ways that result in forming a non-degenerate rectangle. To form a non-degenerate rectangle, we need to have two pairs of positions that are equidistant and parallel to each other on the grid.
There are four possible orientations for the sides of a rectangle: horizontal, vertical, and two diagonals (ascending and descending).
For a horizontal rectangle, each player needs to choose two positions that are in the same row. There are 3 rows to choose from, so the number of ways to do this is (3 choose 2) = 3.
For a vertical rectangle, each player needs to choose two positions that are in the same column. There are 8 columns to choose from, so the number of ways to do this is (8 choose 2) = 28.
For the ascending diagonal rectangle, each player needs to choose two positions that form an ascending diagonal line. There are 2 diagonal lines in each row, so the number of ways to do this is (2 choose 2) = 1 in each row. Since there are 3 rows, the total number of ways is 1^3 = 1.
For the descending diagonal rectangle, each player needs to choose two positions that form a descending diagonal line. Similar to the ascending case, there is 1 diagonal line in each row, so the number of ways is also 1.
Therefore, the total number of ways that result in forming a non-degenerate rectangle is 3 + 28 + 1 + 1 = 33.
Finally, we can calculate the probability by dividing the number of ways to form a non-degenerate rectangle by the total number of possible ways to choose the positions:
Probability = 33 / (24^4)
To express the probability as a fraction in its simplest form, we need to find the greatest common divisor (GCD) of 33 and (24^4) and divide both numerator and denominator by that value.
Calculating this, we find that GCD(33, 24^4) = 1.
Therefore, the probability can be expressed as 33/((24^4)/1) = 33/(24^4) = 33/331776.
Finally, the value of a+b is 33 + 331776 = 331809.