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?

Question

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?

Answer 1

To find the probability that the tokens form the vertices of a non-degenerate rectangle, we need to consider all possible combinations of positions that could form a rectangle.

First, let's determine the total number of possible positions on the 3×8 grid. Since there are 3 rows and 8 columns, the total number of positions is 3 × 8 = 24.

Now, let's consider the different possibilities for a rectangle. A rectangle can be formed by selecting two opposite corners (vertices). There are 24 positions to choose from for the first vertex. After choosing the first vertex, there are 23 positions remaining for the second vertex, but we need to subtract the positions that are not valid for forming a non-degenerate rectangle.

Here's the breakdown of the counting:

- For the second vertex to form a non-degenerate rectangle, it must be in the same row or column as the first vertex. There are 7 remaining positions in the chosen row (excluding the position of the first vertex) and 2 remaining rows (excluding the row of the first vertex). So, there are (7 + 2) = 9 valid positions for the second vertex.
- Once we have chosen the first two vertices, the third vertex must be in the same row or column as one of the first two vertices. Following similar reasoning, there are 7 valid positions for the third vertex.
- Finally, the fourth vertex must be in the same row or column as one of the three already chosen vertices. Again, there are 7 valid positions.

Multiplying the number of choices together for each vertex, we get: 24 × 9 × 7 × 7.

Therefore, the total number of valid combinations for forming a rectangle is 24 × 9 × 7 × 7.

To calculate the probability, we divide the number of valid combinations by the total number of possible positions on the grid, which is 24.

So, the probability is (24 × 9 × 7 × 7) / 24 = 63.

Therefore, the probability a/b is 63/1.

The value of a+b is 63+1 = 64.