Question

Five persons entered the lift cabin on the ground floor of the 8-floor house. Suppose that each of them independently and with equal probability, can leave the cabin at any floor beginning with the first. The probability of all five persons leaving at different floors is
A) \[\dfrac{{{}^8{P_5}}}{{{7^4}}}\]
B) \[\dfrac{{{}^9{P_5}}}{{{7^6}}}\]
C) \[\dfrac{{{}^7{P_5}}}{{{7^5}}}\]
D) \[None\,\,of\,these\]

Answer 1

Hint: First we have to find total outcomes and then favourable outcomes using permutations and combinations. Then using formula we get the required solution.
The formula of the probability of an event is:
\[P(A) = \dfrac{{Number\,of\,Favourable\,outcome\,}}{{Total\,Number\,of\,Favourable\,outcomes}}\]
Combination formula
\[{}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}\]
\[{}^n{C_r}\]= number of the combination.
\[n\]=total number of objects present in the set.
\[r\]=number of chosen objects.
Permutation formula
\[{}^n{P_r} = \dfrac{{n!}}{{(n - r)!}}\]
\[{}^n{P_r}\]=permutation
\[n\]=total number of objects
\[r\]=number of objects selected

Complete step by step answer:
There are in total \[8\] floors,
 So except the ground floor, there are seven floors.
So, If all five-person entered the lift on the ground floor, then any of them can independently leave the cabin on any floor.
So every person will have \[7\] options since there are \[7\] floors beside the ground floor.
Since there are five person, they can leave the cabin at any of the floors
\[
  7 \times 7 \times 7 \times 7 \times 7 = {7^5} \\
  5 \\
 \]
So, the total number of ways are: \[{7^5}\]
And,

the favorable number of ways, that is, the number of ways, in which \[5\] persons leave at different floors is given by
= the number of ways, in which \[5\] persons leave at different floors \[ \times \]arrangement of \[5\] person
\[
   = ({}^7{C_5} \times 5!) \\
   = (\dfrac{{7!}}{{2!}}) \\
 \]
We can also write it as:
\[ = {}^7{P_5}\]

∴ The required probability
\[ = \dfrac{{favourable{\text{ }}number{\text{ }}of{\text{ }}ways}}{{{\text{total }}number{\text{ }}of{\text{ }}ways,}}\]
= \[\dfrac{{{}^7{P_5}}}{{{7^5}}}\]

So, the correct answer is “Option C”.

Note:
All five-person entered the lift on the ground floor, then any of them can independently leave the cabin on any floor, so the total number of ways in which they can leave is the multiplication of all choices each person have and since there is only five-person then there are at most five different floors in which person may leave the cabin.