XAT 2023 — QA & DI Question 11
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?
Answer & solution
- B
- C
- D
- E
Required probability =
No. of ways of selecting 2 smaller squares = 64C2 = 32 × 63
Now, to select 2 squares with common side, there are two cases possible.
Case 1: Horizontal pair is selected.
In the first row 2 squares can be selected in 7 ways i.e., (R1C1, R2C2), (R1C2, R2C3), ..., (R1C7, R2C8)
Similarly 2 squares can be selected from each row in 7 ways.
∴ No. of ways of selecting 2 smaller horizontal pair of squares = 7 × 8 = 56.
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Case 2: Vertical pair is selected.
In the first column 2 squares can be selected in 7 ways i.e., (C1R1, C1R2), (C1R2, C1R3), ..., (C1R7, C1R8)
Similarly 2 squares can be selected from each column in 7 ways.
∴ No. of ways of selecting 2 smaller vertical pair of squares = 7 × 8 = 56.
⇒ Total no. of ways of selecting 2 smaller squares with common side = 56 + 56 = 112
∴ Required probability = =
Hence, option (a).