A box contains 20 balls numbered 1 to 20. A ball is drawn at r... | Mathematics MCQ | Examoogle

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A box contains 20 balls numbered 1 to 20. A ball is drawn at random. The probability that the number is a multiple of 3 or 4 is:

1/2

2/5

3/5

9/20

Correct Answer

⚙️ TE • Technical Inclusion-Exclusion PrincipleMathematicsProbability

Option D

9/20

Quick Summary:

Count multiples of 3 (20/3 = 6) and 4 (20/4 = 5), then subtract common multiples of 12 (20/12 = 1). Total is (6 + 5 - 1) / 20 = 10/20.

📐MAMath SolutionInclusion-Exclusion Principle

📋 Given

Total balls = 20 (numbered 1 to 20). Events: multiples of 3 and multiples of 4.

🔢 Formula Used

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)P(A∪B)=P(A)+P(B)−P(A∩B)

⚡ Exam Hall Shortcut / Speed Trick

Count multiples of 3 (20/3 = 6) and 4 (20/4 = 5), then subtract common multiples of 12 (20/12 = 1). Total is (6 + 5 - 1) / 20 = 10/20.

⚠️ Common Student Trap / Pitfall

Students often double-count the number 12 (multiple of both 3 and 4), leading to an incorrect count of 11 instead of 10.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

List multiples of 3

Identify numbers from 1 to 20 divisible by 3: {3, 6, 9, 12, 15, 18}. There are 6 such numbers.

n(A)=6n(A) = 6n(A)=6

List multiples of 4

Identify numbers from 1 to 20 divisible by 4: {4, 8, 12, 16, 20}. There are 5 such numbers.

n(B)=5n(B) = 5n(B)=5

Identify intersection

Numbers divisible by both 3 and 4 are multiples of their LCM, which is 12. Only {12} is in the range.

n(A∩B)=1n(A \cap B) = 1n(A∩B)=1

Apply Inclusion-Exclusion Principle

Find the count of numbers that are multiples of 3 or 4: $n(A \cup B) = n(A) + n(B) - n(A \cap B) = 6 + 5 - 1 = 10$ .

n(A∪B)=10n(A \cup B) = 10n(A∪B)=10

Calculate final probability

The probability is the favorable outcomes divided by the total number of outcomes: $P = \frac{10}{20} = \frac{1}{2}$ .

P=1020=12P = \frac{10}{20} = \frac{1}{2}P=2010​=21​

✅

A is correct because the count of favorable numbers is 10 out of 20, which simplifies to 1/2.

Core Concepts Used

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💡 EXAM TIP

This set theory concept is frequently used in Permutations and Combinations to avoid double-counting in arrangements and selections.

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