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 calculate the probability that two unit squares chosen from the 4 x 4 grid do not have a vertex in common, we need to determine the total number of possible pairs of squares and the number of pairs that have no common vertices.
Step 1: Finding the Total Number of Possible Pairs
Since there are 16 unit squares in the 4 x 4 grid, we want to calculate the total number of combinations of choosing two squares from these 16 squares, which can be found using the combination formula.
The combination formula is given by:
nCr = n! / (r!(n-r)!)
Where n is the total number of items and r is the number of items being chosen.
In this case, we have n = 16 and r = 2. Plugging these values into the formula:
16C2 = 16! / (2!(16-2)!) = 16! / (2!14!) = (16 * 15) / (2 * 1) = 120
So, there are 120 possible pairs of squares that can be chosen.
Step 2: Finding the Number of Pairs with No Common Vertices
To count the number of pairs of squares that do not have any common vertices, we need to consider a few cases.
Case 1: Two squares chosen from different rows and different columns
We can select any square from one row and any square from another row, and similarly, any square from one column and any square from another column. Since there are 4 rows and 4 columns, the number of pairs for this case is 4 * 4 = 16.
Case 2: Two squares chosen from the same row or column
We can select any square from a row and any square from that same row, but excluding the square itself. Similarly, we can select any square from a column, and any square from that same column, but excluding the square itself.
For each row or column, there are 4 squares to choose from, so the number of pairs for this case is (4 * 3) + (4 * 3) = 24.
Adding up the pairs from both cases, we have a total of 16 + 24 = 40 pairs of squares that have no common vertices.
Step 3: Calculating the Probability
Finally, to calculate the probability that the two chosen squares do not have a common vertex, we divide the number of pairs with no common vertices (40) by the total number of possible pairs (120).
Probability = P(no common vertices) = 40 / 120 = 1 / 3
Therefore, the probability that two unit squares chosen from the 4 x 4 grid do not have a common vertex is 1/3 or 1 in 3.