Question

Five persons entered the lift cabin on the ground floor of an 8 floor building. If, 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 and (ii) at different floors.

Answer 1

Hint: For leaving at any one of the seven floors there are 7 different ways to leave for each of the five persons. And for leaving at different floors the first person will have 7 different ways but the next person will have one less different way than the first person and similarly it will get one less for consecutive persons.

Complete step-by-step answer:
(i) Suppose there are 5 persons.
The first person can leave the lift cabin in 7 different ways.
The second person can also leave the lift cabin in 7 different ways.
The third person can also leave the lift cabin in 7 different ways.
The fourth person can also leave the lift cabin in 7 different ways.
The fifth person can also leave the lift cabin in 7 different ways.
Thus, by product rule, the total number of ways, in which each of the five persons can leave the lift cabin at any one of the floors $=7\times 7\times 7\times 7\times 7={{7}^{5}}=16807$

(ii) Suppose there are 5 persons.
The first person can leave the lift cabin in 7 different ways.
The second person can also leave the lift cabin in 6 different ways.
The third person can also leave the lift cabin in 5 different ways.
The fourth person can also leave the lift cabin in 4 different ways.
The fifth person can also leave the lift cabin in 3 different ways.
Thus, by product rule, the total number of ways, in which each of the five persons can leave the lift cabin at different floors $=7\times 6\times 5\times 4\times 3=2520$

Note: The fundamental principle of counting (product rule) is a way to figure out the number of outcomes in a probability problem. Basically, we multiply the events together to get the total number of outcomes.