Problem:
The numbers are randomly placed into the squares of a grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and column is odd.
Answer Choices:
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Solution:
The sum of three integers is odd exactly when either all of the integers are odd, or one is odd and two are even. Five of the numbers are odd, so at least one row must contain two or more odd numbers. Thus one row must contain three odd numbers and no even numbers, and the other two rows must contain one odd number and two even numbers. The same is true of the three columns. There are ways to choose which row and which column contain all odd numbers, and then the remaining four squares must have even numbers. There are ways in total to choose which squares have odd numbers and which have even numbers, so the desired probability is .
The problems on this page are the property of the MAA's American Mathematics Competitions