Question

Five persons entered the lift cabin on the ground floor of an 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 that all five persons leaving at different floors is $\frac{{{}^{7}P}_k}{7^5}$.find the value of $k$

Answer 1

Any person out of the $5$ people can leave at any of the $7$ floors above. Hence, total possibilities$=7\times 7\times 7\times 7\times 7=7^5$
If all the $5$ people have to be at different floors,then these $5$ can be arranged on $5$ different floors out of the $7$ floors in ${}_7{P}_5$ ways.
Hence, probability $=\frac{{}_7{P}_5}{7^5}$
So, $k=5$