<p>Five persons entered the lift cabin on the ground floor of an 8-floor building. Suppose each of them can leave the cabin independently at any floor beginning with the first. Find the total number of ways in which each of the five persons can leave the cabin <br>(a) at any one of the 7 floors<br>(b) at different floors.</p>
Step-by-Step Solution
Key Concept: For part (a), each of 5 persons independently chooses from 7 floors, giving 7^5 ways. For part (b), we need ordered selections without repetition: the first person has 7 choices, second has 6, third has 5, etc., which is P(7,5) = 7!/(7-5)! = 2520.
<p><strong>Step 1 (Part a):</strong> Each of the 5 persons can independently leave at any of the 7 floors (floors 1-7; they don't leave at ground floor). By the multiplication principle, the total number of ways = 7 × 7 × 7 × 7 × 7 = <strong>7^5 = 16,807</strong></p><p><strong>Step 2 (Part b):</strong> Now each person must leave at a different floor. This is a permutation problem:</p><ul><li>Person 1: can choose any of 7 floors</li><li>Person 2: can choose any of remaining 6 floors</li><li>Person 3: can choose any of remaining 5 floors</li><li>Person 4: can choose any of remaining 4 floors</li><li>Person 5: can choose any of remaining 3 floors</li></ul><p>Total ways = 7 × 6 × 5 × 4 × 3 = P(7,5) = 7!/2! = <strong>2520</strong></p><p>∴ Answer: (a) 7^5 = 16,807, (b) 2520</p>
Correct Answer: (a) 7^5, (b) 2520