Q97 / 180 · COMEDK 2025 (10 May Shift 2) · Mathematics
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$\dfrac{7}{{ }^7 P_5}$
- Correct answerB$\dfrac{{ }^7 P_5}{7^5}$
- C$\dfrac{{ }^7 P_5}{5^7}$
- D$\dfrac{5!}{7^5}$
Solution
There are 8 floors in the apartment, but the persons leave at any floor from the first to the eighth floor. Thus, there are 7 possible floors for each person to leave (1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th floor).
Each of the 5 persons can choose any of the 7 floors independently. Therefore, the total number of ways for the 5 persons to leave the cabin is $7 \times 7 \times 7 \times 7 \times 7 = 7^5$.
For all 5 persons to leave at different floors, we need to select 5 distinct floors out of the 7 available floors and assign them to the 5 persons. The number of ways to do this is the number of permutations of 7 items taken 5 at a time, which is $^{7}P_{5}$.
The probability that all 5 persons leave at different floors is the ratio of the number of favorable outcomes to the total number of outcomes, which is $\dfrac{^{7}P_{5}}{7^5}$.
Answer: $\dfrac{{ }^7 P_5}{7^5}$
Each of the 5 persons can choose any of the 7 floors independently. Therefore, the total number of ways for the 5 persons to leave the cabin is $7 \times 7 \times 7 \times 7 \times 7 = 7^5$.
For all 5 persons to leave at different floors, we need to select 5 distinct floors out of the 7 available floors and assign them to the 5 persons. The number of ways to do this is the number of permutations of 7 items taken 5 at a time, which is $^{7}P_{5}$.
The probability that all 5 persons leave at different floors is the ratio of the number of favorable outcomes to the total number of outcomes, which is $\dfrac{^{7}P_{5}}{7^5}$.
Answer: $\dfrac{{ }^7 P_5}{7^5}$
Hence, the correct answer is option: 'B'.