Permutations & Combinations
Fundamental Principle of Counting
Grade 11
Question:
<p>Four buses run between Bhopal and Gwalior. If a man goes from Gwalior to Bhopal by a bus and comes back to Gwalior by another bus, find the total possible ways.</p>
Step-by-Step Solution
Key Concept: The outbound and return journeys are independent events—we can choose any of 4 buses to go AND any of the remaining 3 buses to return (since he must use a different bus coming back).
<p><strong>Step 1:</strong> Identify the two independent events: (i) Going from Gwalior to Bhopal, (ii) Returning from Bhopal to Gwalior by a <u>different</u> bus.</p><p><strong>Step 2:</strong> For the outbound journey: He can choose any of the 4 buses = <strong>4 ways</strong>.</p><p><strong>Step 3:</strong> For the return journey: He must use a <u>different bus</u>, so he can choose from the remaining 3 buses = <strong>3 ways</strong>.</p><p><strong>Step 4:</strong> By the Fundamental Counting Principle, total ways = 4 × 3 = <strong>12</strong>.</p><p>∴ <strong>Answer: 12</strong></p>
Correct Answer: 12