Find the probability that a four-digit number comprising the digits $2, 5, 6$ and $7$ would be divisible by $4$.

The correct answer is Option b: $1/3$. **Probability of a 4-Digit Number Being Divisible by 4** Given digits: $2, 5, 6, 7$ (no digit is repeated). **Step 1: Total possible 4-digit numbers** The number of permutations of 4 distinct digits is: $$\text{Total outcomes} = 4! = 24$$ **Step 2: Divisibility by 4 rule** A number is divisible by 4 if the last two digits form a number divisible by 4. Let's find the favorable two-digit combinations from $\{2, 5, 6, 7\}$: - $52$ ($52 = 4 \times 13$, Yes) - $56$ ($56 = 4 \times 14$, Yes) - $72$ ($72 = 4 \times 18$, Yes) - $76$ ($76 = 4 \times 19$, Yes) There are 4 favorable last-two-digit pairs. **Step 3: Calculating favorable outcomes** For each pair, the remaining 2 digits can be arranged in the first two positions in $2! = 2$ ways. $$\text{Favorable outcomes} = 4 \times 2 = 8$$ **Step 4: Probability calculation** $$P = \frac{8}{24} = \frac{1}{3}$$ Hence, **Option B** is the correct answer.

ProbabilityMCQMTP May 20Question 3352 of 187

All Questions

Find the probability that a four-digit number comprising the digits $\displaystyle 2, 5, 6$ and $\displaystyle 7$ would be divisible by $\displaystyle 4$ .

Options

\displaystyle 1/4

\displaystyle 1/3

\displaystyle 1/2

\displaystyle 1

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Correct Answer

✅ Option b — $\displaystyle 1/3$

All Options:

A $\displaystyle 1/4$
B $\displaystyle 1/3$
C $\displaystyle 1/2$
D $\displaystyle 1$

Detailed Solution & Explanation

**Probability of a 4-Digit Number Being Divisible by 4** Given digits:

\displaystyle 2, 5, 6, 7

(no digit is repeated). **Step 1: Total possible 4-digit numbers** The number of permutations of 4 distinct digits is:

\text{Total outcomes} = 4! = 24

**Step 2: Divisibility by 4 rule** A number is divisible by 4 if the last two digits form a number divisible by 4. Let's find the favorable two-digit combinations from

\displaystyle \{2, 5, 6, 7\}

: -

\displaystyle 52

(

\displaystyle 52 = 4 \times 13

, Yes) -

\displaystyle 56

(

\displaystyle 56 = 4 \times 14

, Yes) -

\displaystyle 72

(

\displaystyle 72 = 4 \times 18

, Yes) -

\displaystyle 76

(

\displaystyle 76 = 4 \times 19

, Yes) There are 4 favorable last-two-digit pairs. **Step 3: Calculating favorable outcomes** For each pair, the remaining 2 digits can be arranged in the first two positions in

\displaystyle 2! = 2

ways.

\text{Favorable outcomes} = 4 \times 2 = 8

**Step 4: Probability calculation**

P = \frac{8}{24} = \frac{1}{3}

Hence, **Option B** 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.

Key Concepts to Understand

Probability

A numerical measure of the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). Formula: P(A) = Number of favorable outcomes / Total number of equally likely outcomes.

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Find the probability that a four-digit number comprising the digits 2,5,6\displaystyle 2, 5, 62,5,6 and 7\displaystyle 77 would be divisible by 4\displaystyle 44.

Options

Correct Answer

All Options:

Detailed Solution & Explanation

About This Chapter: Probability

Key Concepts to Understand

Probability

More Questions from Probability

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Find the probability that a four-digit number comprising the digits $\displaystyle 2, 5, 6$ and $\displaystyle 7$ would be divisible by $\displaystyle 4$ .