The probability that a four digit number comprising the digits 2, 5, 6 and 7 without repetition of digits, would be divisible by 4 is

Option d: 1/3. Solution: Probability of a 4-Digit Number Being Divisible by 4 Given digits: 2, 5, 6, 7 (to be used without repetition to form a 4-digit number). Step 1: Total possible 4-digit numbers The number of permutations of 4 distinct digits is: Total outcomes = 4! = 24 Step 2: Divisibility by 4 rule A number is divisible by 4 if and only if the number formed by its last two digits is divisible by 4. Let's list all possible 2-digit combinations from 2, 5, 6, 7 and check their divisibility by 4: - 25 (No) - 26 (No) - 27 (No) - 52 ( 52 = 4 × 13 , Yes) - 56 ( 56 = 4 × 14 , Yes) - 57 (No) - 62 (No) - 65 (No) - 67 (No) - 72 ( 72 = 4 × 18 , Yes) - 75 (No) - 76 ( 76 = 4 × 19 , Yes) The favorable last two digits are: 52, 56, 72, 76 (4 possibilities). Step 3: Calculating favorable permutations For each of the 4 favorable pairs of last two digits, the remaining 2 digits can be arranged in the first two positions in: 2! = 2 ways Thus, the total number of favorable 4-digit numbers is: Favorable outcomes = 4 × 2 = 8 Step 4: Probability calculation P(divisible by 4) = outcomesTotal outcomes = 8/24 = 1/3 Hence, Option D is the correct answer.

The probability that a four digit number comprising the digits $2, 5, 6$ and $7$ without repetition of digits, would be divisible by $4$ is

The correct answer is Option d: $1/3$. **Probability of a 4-Digit Number Being Divisible by 4** Given digits: $2, 5, 6, 7$ (to be used without repetition to form a 4-digit number). **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 and only if the number formed by its last two digits is divisible by 4. Let's list all possible 2-digit combinations from $\{2, 5, 6, 7\}$ and check their divisibility by 4: - $25$ (No) - $26$ (No) - $27$ (No) - $52$ ($52 = 4 \times 13$, Yes) - $56$ ($56 = 4 \times 14$, Yes) - $57$ (No) - $62$ (No) - $65$ (No) - $67$ (No) - $72$ ($72 = 4 \times 18$, Yes) - $75$ (No) - $76$ ($76 = 4 \times 19$, Yes) The favorable last two digits are: $52, 56, 72, 76$ (4 possibilities). **Step 3: Calculating favorable permutations** For each of the 4 favorable pairs of last two digits, the remaining 2 digits can be arranged in the first two positions in: $$2! = 2 \text{ ways}$$ Thus, the total number of favorable 4-digit numbers is: $$\text{Favorable outcomes} = 4 \times 2 = 8$$ **Step 4: Probability calculation** $$P(\text{divisible by 4}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{8}{24} = \frac{1}{3}$$ Hence, **Option D** is the correct answer.

ProbabilityMCQPYQ June 23Question 3335 of 187

All Questions

The probability that a four digit number comprising the digits $\displaystyle 2, 5, 6$ and $\displaystyle 7$ without repetition of digits, would be divisible by $\displaystyle 4$ is

Options

\displaystyle 1/2

\displaystyle 3/4

\displaystyle 1/4

\displaystyle 1/3

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Correct Answer

✅ Option d — $\displaystyle 1/3$

All Options:

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

Detailed Solution & Explanation

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

\displaystyle 2, 5, 6, 7

(to be used without repetition to form a 4-digit number). **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 and only if the number formed by its last two digits is divisible by 4. Let's list all possible 2-digit combinations from

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

and check their divisibility by 4: -

\displaystyle 25

(No) -

\displaystyle 26

(No) -

\displaystyle 27

(No) -

\displaystyle 52

(

\displaystyle 52 = 4 \times 13

, Yes) -

\displaystyle 56

(

\displaystyle 56 = 4 \times 14

, Yes) -

\displaystyle 57

(No) -

\displaystyle 62

(No) -

\displaystyle 65

(No) -

\displaystyle 67

(No) -

\displaystyle 72

(

\displaystyle 72 = 4 \times 18

, Yes) -

\displaystyle 75

(No) -

\displaystyle 76

(

\displaystyle 76 = 4 \times 19

, Yes) The favorable last two digits are:

\displaystyle 52, 56, 72, 76

(4 possibilities). **Step 3: Calculating favorable permutations** For each of the 4 favorable pairs of last two digits, the remaining 2 digits can be arranged in the first two positions in:

2! = 2 \text{ ways}

Thus, the total number of favorable 4-digit numbers is:

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

**Step 4: Probability calculation**

P(\text{divisible by 4}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{8}{24} = \frac{1}{3}

Hence, **Option D** 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.

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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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The probability that a four digit number comprising the digits 2,5,6\displaystyle 2, 5, 62,5,6 and 7\displaystyle 77 without repetition of digits, would be divisible by 4\displaystyle 44 is

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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The probability that a four digit number comprising the digits $\displaystyle 2, 5, 6$ and $\displaystyle 7$ without repetition of digits, would be divisible by $\displaystyle 4$ is