Question

An elevator starts with $m$ passengers and stops at $n$ floors $(m\le n)$. The probability that no two passengers alight at the same floor is

• A. $\frac{{}^n{P}_m}{m^n}$
• B. $\frac{{}^n{P}_m}{n^m}$
• C. $\frac{{}^n{C}_m}{m^n}$
• D. $\frac{{}^n{C}_m}{n^m}$

Answer 1

The correct option is B $\frac{{}^n{P}_m}{n^m}$

A person can alight at any one of $n$ floors.

Therefore, the number of ways in which $m$ passengers can alight at $n$ floors is $n\times n\times n\times ...\times n(m$-times$)=n^m.$

The number of ways in which all passengers can alight at different floors is ${}^n{C}_m\times m!={}^n{P}_m.$

Hence, required probability $=\frac{{}^n{P}_m}{n^m}.$