There is a 4 x 4 square grid. Two unit squares are chosen from the 16 unit squares possible. What is the probability that they do NOT have a vertex in common? Express your answer as a common fraction.
To find the probability that two unit squares chosen from a 4x4 grid do not have a vertex in common, we need to first count the total number of possible combinations of choosing two squares and then determine the number of combinations where the squares do not share a vertex.
Step 1: Counting the total number of combinations
There are a total of 16 unit squares in the 4x4 grid, and we need to choose 2 squares out of them. We can use the formula for combinations to calculate this:
Total combinations = C(16, 2) = 16! / (2! * (16-2)!)
= (16 * 15) / (2 * 1)
= 240
So, there are 240 possible combinations of choosing two squares from the grid.
Step 2: Counting the combinations without a common vertex
To find the number of combinations where the two squares do not share a vertex, we can first choose the first square, and then choose the second square in a way that avoids sharing a vertex with the first square.
Let's consider the first square chosen - it can be any of the 16 unit squares. When we choose this square, it eliminates its adjacent squares (4 for corner squares, 6 for edge squares, and 9 for the center squares) as options for the second square.
For example, if we choose a corner square as the first square, it eliminates 4 adjacent squares, leaving 12 options for the second square. Alternatively, if we choose an edge square as the first square, it eliminates 6 adjacent squares, leaving 10 options for the second square. Finally, if we choose a center square as the first square, it eliminates 9 adjacent squares, leaving 7 options for the second square.
Since there are 4 corner squares, 8 edge squares, and 4 center squares in the 4x4 grid, we need to sum up the possibilities for each case.
Number of combinations without a common vertex:
= 4 * 12 + 8 * 10 + 4 * 7
= 48 + 80 + 28
= 156
Step 3: Calculating the probability
The probability of the two squares not having a common vertex is given by the number of combinations without a common vertex divided by the total number of combinations:
Probability = Number of combinations without a common vertex / Total combinations
Substituting the values, we have:
Probability = 156 / 240
So, the probability that two unit squares chosen from the 4x4 grid do not have a vertex in common is 13/20 (expressed as a common fraction).