Question:
Out of the first $20$ consecutive natural numbers, $3$ numbers are chosen at random. If these $3$ numbers are in arithmetic progression with common difference $d\in\mathbb N$, then the probability of getting those $3$ numbers whose common difference is a prime number is
Out of the first $20$ consecutive natural numbers, $3$ numbers are chosen at random. If these $3$ numbers are in arithmetic progression with common difference $d\in\mathbb N$, then the probability of getting those $3$ numbers whose common difference is a prime number is
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For a three-term A.P., count possible starting terms using the condition $a+2d\le n$.
Updated On: Jun 3, 2026
- $\dfrac{4}{9}$
- $\dfrac{1}{3}$
- $\dfrac{23}{45}$
- $\dfrac{29}{90}$
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The Correct Option is C
Solution and Explanation
Step 1: Concept
Count all possible three-term arithmetic progressions and then count those having a prime common difference.
Step 2: Meaning
A three-term A.P. is of the form \[ a,\ a+d,\ a+2d. \] Since all terms must belong to $\{1,2,\ldots,20\}$, \[ a+2d\le 20. \]
Step 3: Analysis
Total number of three-term A.P.s: \[ \sum_{d=1}^{9}(20-2d) = 18+16+14+12+10+8+6+4+2 = 90. \] Prime values of $d$ are \[ 2,3,5,7. \] Corresponding numbers of A.P.s are \[ 16,\ 14,\ 10,\ 6. \] Hence favorable cases \[ 16+14+10+6=46. \] Therefore \[ P=\frac{46}{90} =\frac{23}{45}. \]
Step 4: Conclusion
Thus the required probability is \[ \frac{23}{45}. \]
Final Answer: (C)
Count all possible three-term arithmetic progressions and then count those having a prime common difference.
Step 2: Meaning
A three-term A.P. is of the form \[ a,\ a+d,\ a+2d. \] Since all terms must belong to $\{1,2,\ldots,20\}$, \[ a+2d\le 20. \]
Step 3: Analysis
Total number of three-term A.P.s: \[ \sum_{d=1}^{9}(20-2d) = 18+16+14+12+10+8+6+4+2 = 90. \] Prime values of $d$ are \[ 2,3,5,7. \] Corresponding numbers of A.P.s are \[ 16,\ 14,\ 10,\ 6. \] Hence favorable cases \[ 16+14+10+6=46. \] Therefore \[ P=\frac{46}{90} =\frac{23}{45}. \]
Step 4: Conclusion
Thus the required probability is \[ \frac{23}{45}. \]
Final Answer: (C)
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