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Five persons entered the lift cabin on t...

Five persons entered the lift cabin on the ground floor of an 8-floor house. 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 (i) at any one of the 7 floors (ii) at different floors.

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The correct Answer is:
(a)`7^(5)` (b) 2520
Let `A_(1),A_(2),A_(3),A_(4),A_(5)` are five persons.
(a) `A_(1)` can leave the cabin at any of the seven floors. So, `A_(1)` can leave the cabin in 7 ways. Similarly, each of `A_(2),A_(3),A_(4),A_(5)` can leave the cabin in 7 ways. Thus, the total number of ways in which each of the five persons can leave the cabin at any of the seven floors is `7xx7xx7xx7xx7=7^(5)`
(b)` A_(1)` can leave the cabin at any of the seven floors. So, `A_(1)` can leave the cabin in 7 ways. Now, `A_(2)` can leave the cabin at any of the remaining 6 floors. So, `A_(2)` can leave the cabin in 6 ways. Similarly, `A_(3),A_(4) " and" A_(5)` can leave the cabin in 5, 4 and 3 ways respectively. Thus, the total number of ways in which each of the five persons can leave the cabin at different floors is `7xx6xx5xx4xx3=2520`
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