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Grid Board Adjacent Squares Probability

XAT 2023 · Quantitative Ability & Data Interpretation · Medium · Probability

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A painter draws 64 equal squares of 1 square inch on a square canvas measuring 64 square inches. She chooses two squares (1 square inch each) randomly and then paints them. What is the probability that two painted squares have a common side?

  1. A.

    1122016\frac{112}{2016}

  2. B.

    13\frac{1}{3}

  3. C.

    512100034\frac{512}{100034}

  4. D.

    397\frac{3}{97}

  5. E.

    7108\frac{7}{108}

Answer

A

Explanation

Grid is 8×88 \times 8. Total pairs of squares = (642)=64×632=2016\binom{64}{2} = \frac{64 \times 63}{2} = 2016. Horizontal adjacent pairs per row = 7, total across 8 rows = 7×8=567 \times 8 = 56. Vertical adjacent pairs per column = 7, total across 8 columns = 7×8=567 \times 8 = 56. Total favorable pairs = 56+56=11256 + 56 = 112. Probability = 1122016\frac{112}{2016}.

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