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GATE DA 2026 Q64: Row and column sums of a random 0/1 matrix

Question

Let A5×5A_{5 \times 5} be a matrix such that each of its elements follows Bernoulli(p=0.50)Bernoulli(p = 0.50) distribution independently.

The probability that the row-sum of the second row and the column-sum of the third column are both equal to 3 is ________. (Rounded off to two decimal places)

Try it: numerical answer: type the value

Numerical answer. No negative marking.
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Answer: 0.08 to 0.12 (official GATE 2026 final answer key)

Concept used: Condition on the shared entry a23a_{23}; the other four entries of the row and of the column are independent.

First step: If a23=1a_{23} = 1 each needs 2 of 4 others ((42)/16\binom{4}{2}/16); if a23=0a_{23} = 0 each needs 3 of 4 ((43)/16\binom{4}{3}/16).

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Source: official GATE 2026 DA question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key

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