3x3 Matrix: Even Sums
The numbers 1 through 9 are randomly assigned to the 9 cells of a \(3\times3\) matrix, with each number appearing exactly once. What is the probability that the sum of the numbers in each row and each column is even?
Answer
Note that among 1 through 9 there are 5 odd numbers and 4 even numbers. A row or column sum is even if and only if it contains an even number of odd entries (i.e. 0 or 2 in a row of 3). However, since the total number of odd numbers in the matrix is 5 (an odd number), it is impossible for every row and column to have an even number of odd entries. Hence, the probability is \(0\).