Question

There are four hotels in a town. If $3$ men check into the hotels in a day, then what is the probability that each checks into a different hotel?

• A. $\frac{6}{7}$
• B. $\frac{1}{8}$
• C. $\frac{3}{8}$
• D. $\frac{5}{9}$

Answer 1

The correct option is C

$\frac{3}{8}$

Explanation for the correct option:

Finding the probability each one checks into a different hotel:

Three males used a total of four different methods to check into the four hotels

$\begin{array}{rcl}& =& 4\times 4\times 4\\ & =& 4^3\end{array}$

The number of distinct ways each man checks into a hotel is $=4\times 3\times 2$

There are four options for the first person, three for the second, and two for the third.

$\therefore \mathrm{P}\left(\text{Everyone stays at a different hotel}\right)$

$$\begin{gathered} =\frac{4\times 3\times 2}{4\times 4\times 4} \\ =\frac{{\displaystyle 3}}{{\displaystyle 8}} \end{gathered}$$

Hence, The correct option is C.