Question:
If five unit squares are selected at random from a chess board, then the probability that they all lie on a diagonal is
If five unit squares are selected at random from a chess board, then the probability that they all lie on a diagonal is
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For probability on a chessboard, count favorable diagonal arrangements first and divide by total selections.
Updated On: Jun 3, 2026
- $\dfrac{112}{{}^{64}C_5}$
- $\dfrac{56}{{}^{64}C_5}$
- $\dfrac{448}{{}^{64}C_5}$
- $\dfrac{224}{{}^{64}C_5}$
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The Correct Option is D
Solution and Explanation
Step 1: Concept
Count favorable selections and divide by total selections.
Step 2: Meaning
Total ways of choosing $5$ squares from $64$ squares: \[ {}^{64}C_5. \]
Step 3: Analysis
On an $8\times8$ chessboard, diagonals of length at least $5$ contribute favorable selections. Counting all possible selections of $5$ squares lying on the same diagonal gives \[ 224 \] favorable cases. Hence \[ P= \frac{224}{{}^{64}C_5}. \]
Step 4: Conclusion
Therefore the required probability is \[ \frac{224}{{}^{64}C_5}. \]
Final Answer: (D)
Count favorable selections and divide by total selections.
Step 2: Meaning
Total ways of choosing $5$ squares from $64$ squares: \[ {}^{64}C_5. \]
Step 3: Analysis
On an $8\times8$ chessboard, diagonals of length at least $5$ contribute favorable selections. Counting all possible selections of $5$ squares lying on the same diagonal gives \[ 224 \] favorable cases. Hence \[ P= \frac{224}{{}^{64}C_5}. \]
Step 4: Conclusion
Therefore the required probability is \[ \frac{224}{{}^{64}C_5}. \]
Final Answer: (D)
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