Question

Five persons entered the lift cabin on the ground floor of an eight-floor apartment. Suppose that each of them independently and with equal probability, can leave the cabin at any floor beginning with the first floor, then the probability of all five persons leaving at different floors is:

• A. \( \frac{7}{8^5} \)
• B. \( \frac{1}{8^5} \)
• C. \( \frac{5}{8^5} \)
• D. \( \frac{7}{5^3} \)

Hint:

For probability problems, use favorable outcomes divided by total possible outcomes.

Answer 1

The Correct Option is B

Step 1: Total possibilities for each person.
Each person has the choice to leave the lift on any floor between 1 and 8, making the total number of possibilities for each person 8. So, the total possibilities for five persons is \( 8^5 \).

Step 2: Calculating the favorable outcomes.
For the five persons to leave at different floors, the first person has 8 options, the second has 7 options, and so on. The favorable outcomes for all five persons leaving on different floors are \( 8 \times 7 \times 6 \times 5 \times 4 \).

Step 3: Probability calculation.
The probability is the ratio of favorable outcomes to total outcomes: \[ P = \frac{8 \times 7 \times 6 \times 5 \times 4}{8^5} = \frac{1}{8^5}. \]

Step 4: Conclusion.
Thus, the probability of all five persons leaving at different floors is \( \frac{1}{8^5} \), corresponding to option (2).