Question

5 persons entered the lift cabin on the ground floor of an 8 floor building. Suppose that each of them independently and with equal probability, can leave the cabin at any other floor, starting from the first. The probability that all 5 persons leave at different floors is

• A. ${\left(\frac{5}{8}\right)}^5$
• B. $\frac{{}^8{C}_5}{8^5}$
• C. $\frac{5!}{8!}$
• D. $\frac{{}^8{C}_{5}5!}{8^5}$

Answer 1

The correct option is D $\frac{{}^8{C}_{5}5!}{8^5}$

Given, $5$ persons entered the lift cabin at the ground floor of $8$ floor building.
The total number of permutations that $5$ persons can get down in $8$ floor builidng= $8\ast 8\ast 8\ast 8\ast 8=8^5$ (Since each person has 8 chances of getting down)
Total number of permutations such that each get down at different floors${=}^8{P}_5$
$\therefore$ The Required probability$=\frac{{}^8{P}_5}{8^5}=\frac{{(}^8{C}_5\ast 5!)}{8^5}$