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?
Note:
-Players are allowed to have selected the same squares.
-There is no restriction on their choices.
-A degenerate rectangle has 0 area.
-Squares are rectangles.
To solve this problem, we can break it down into smaller steps:
Step 1: Determine the total number of ways the players can choose positions on the grid.
Since each player has 24 possible positions to choose from (3×8), the total number of ways they can choose positions is 24^4 = 331,776.
Step 2: Determine the number of non-degenerate rectangles that can be formed on the grid.
To do this, we need to consider the possible horizontal and vertical lines that can form the sides of the rectangle.
Horizontal lines: There are 3 horizontal lines that can be formed by choosing any 3 rows out of the 8 available. This can be calculated using the binomial coefficient formula: C(8,3) = 56.
Vertical lines: There are 6 vertical lines that can be formed by choosing any 2 columns out of the 3 available. This can be calculated using the binomial coefficient formula: C(3,2) = 3.
The total number of non-degenerate rectangles that can be formed is obtained by multiplying the number of horizontal lines by the number of vertical lines: 56 * 3 = 168.
Step 3: Determine the probability of forming a non-degenerate rectangle.
The probability is given by the ratio of the number of favorable outcomes (forming a non-degenerate rectangle) to the total number of possible outcomes (choosing positions on the grid).
Probability = Number of non-degenerate rectangles / Total number of ways to choose positions = 168 / 331,776.
Step 4: Simplify the probability.
The probability can be further simplified by dividing both the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 168 and 331,776 is 336.
Probability = 168 / 331,776 = (168 ÷ 336) / (331,776 ÷ 336) = 1 / 1,968.
Step 5: Calculate the sum of a and b.
The value of a is 1 and the value of b is 1,968.
a + b = 1 + 1,968 = 1,969.
Therefore, the value of a + b is 1,969.