Question

There are 8 floors on a building including the ground floor. If 4 persons enter the lift in the ground floor, the number of ways in which they can get down the lift if no two persons come out of the lift at the same floor is:
A.${}^7{C_4}$
B.${7^4}$
C.${}^7{P_4}$
D.${4^7}$

Answer 1

Hint: Here we will use the fundamental principle of counting to calculate the number of ways in which 4 persons can come out of the lift with given conditions.

Complete step-by-step answer:
The fundamental principle of counting is a rule which is used to count the total number of possible ways or outcomes in a given situation with certain conditions.
In this you multiply the ways of doing first thing with the ways to do the other thing. For e.g., if there are m ways to perform event a and there are n ways to perform event b, then the total number of ways to perform the simultaneous events a and b are $m \times n$.
In the given question, we have a total of 8 floors including the ground floor.
4 persons enter the lift from the ground floor and we need to calculate the number of ways in which they can get down the lift with a condition that no two should leave at a time.
So, if they entered from the ground floor, there are remaining 7 floors where person 1 can get down.
Therefore, there are 7 ways for person1 to get down the lift.
For person2, excluding the ground floor and the floor where person1 got down, there are 6 floors remaining i.e., person2 can get down in 6 ways.
Similarly, person3 will have 5 ways and person 4 will have 4 ways to get down the lift.
Using the fundamental principle of the counting, we get
The total number of ways in which 4 persons get down without any two at a same floor:$7 \times 6 \times 5 \times 4 = 840$
Now we will check the options.
Option(A), ${}^7{C_4}$= 140. Therefore, option(A) is incorrect.
Option(B), ${7^4}$= 2401. Therefore, option(B) is incorrect.
Option(C), ${}^7{P_4}$= 840. Therefore, option(C) is correct.
Option(D), ${4^7}$= 16384. Therefore, option(D) is incorrect.

Note: Fundamental counting is the best way to solve this type of problem. In this solution, we should check every option so there can be many correct options. One mistake can lead to the wrong answer. Avoid calculation mistakes.