ProbabilityPYQ Jan 26Question 4538 of 187
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Find the probability that a 3-digit number formed using the digits 1, 3, and 5 (without repetition), is divisible by 3?

Options

A1
B0
C13\displaystyle \frac{1}{3}
D23\displaystyle \frac{2}{3}
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Correct Answer

✅ Option a — 1

All Options:

  • A1
  • B0
  • C13\displaystyle \frac{1}{3}
  • D23\displaystyle \frac{2}{3}

Detailed Solution & Explanation

A number is divisible by 3\displaystyle 3 if and only if the sum of its digits is divisible by 3\displaystyle 3.

We are forming a 3\displaystyle 3-digit number using the digits 1\displaystyle 1, 3\displaystyle 3, and 5\displaystyle 5 without repetition.
For any such number, the sum of the digits is:
Sum of digits=1+3+5=9\text{Sum of digits} = 1 + 3 + 5 = 9

Since 9\displaystyle 9 is divisible by 3\displaystyle 3 (9÷3=3\displaystyle 9 \div 3 = 3), every 3\displaystyle 3-digit number formed using these three digits (namely 135\displaystyle 135, 153\displaystyle 153, 315\displaystyle 315, 351\displaystyle 351, 513\displaystyle 513, and 531\displaystyle 531) will be divisible by 3\displaystyle 3.

Thus, the number of favorable outcomes is equal to the total number of possible outcomes (3!=6\displaystyle 3! = 6). The probability is:
Probability=Favorable OutcomesTotal Outcomes=66=1\text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{6}{6} = 1

Hence, **Option A** is the correct answer.

About This Chapter: Probability

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Probability Operations, Expected Value

A logic-heavy chapter dealing with random experiments, events (mutually exclusive, exhaustive), set theory probability, conditional probability, and Bayes' Theorem. It forms the basis for Theoretical Distributions.

View Official ICAI Syllabus

Exam Strategy Tip

Always draw a quick Venn Diagram or tree when faced with 'At least one' or 'Only A but not B' wording. It saves you from double-counting.

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