Odd row and column sums in a random 3 by 3 grid

Question

The numbers $1,2,\dots,9$ are placed in a $3\times3$ grid in a uniformly random arrangement, one per cell. What is the probability that all three row sums and all three column sums are odd?

Short answer

$\tfrac1{14}$ . Only the positions of the 5 odd numbers matter, and a line of three has an odd sum iff it contains 1 or 3 odds. With 5 odds this forces one full row plus one full column of odds (a cross), giving $3\times3=9$ valid patterns out of $\binom95=126$ , i.e. $\tfrac1{14}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Counting and Combinatorics
Series and the Geometric Series: Partial Sums and Convergence
Sample Spaces, Events, and the Probability Axioms

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