Let's solve the question step by step:
We have 5 persons who entered the lift cabin on the ground floor of an 8-floor building. The building has floors 1 to 8, and each person can independently leave at any of the 7 floors (i.e., floors 1 to 7, as the ground floor is where they start).
Part (i) Each person can leave at any one of the 7 floors:
In this case, each person has 7 options (floor 1 to floor 7) to leave the lift. Since there are 5 persons, and each person can choose one of the 7 floors independently, the total number of ways in which the 5 persons can leave the cabin is calculated as:
Total number of ways = 7 × 7 × 7 × 7 × 7 = 7^5
Now, let's calculate 7^5:
7^5 = 16807
Thus, the total number of ways in which each of the five persons can leave the cabin at any one of the 7 floors is 16,807.
Part (ii) Each person must leave at a different floor:
In this case, each of the 5 persons must leave at a different floor, and the 5 floors that they will leave at must be selected from the 7 available floors.
First, we select 5 floors from the 7 available floors. The number of ways to select 5 floors from 7 is given by the combination formula:
Number of ways to choose 5 floors from 7 = C(7, 5) = 7! / (5! × (7-5)!) = (7 × 6) / (2 × 1) = 21
After selecting the 5 floors, we can assign each of the 5 persons to one of the 5 chosen floors. The number of ways to assign 5 persons to 5 floors is simply the number of permutations of 5 persons, which is 5! (5 factorial):
5! = 5 × 4 × 3 × 2 × 1 = 120
Therefore, the total number of ways in which each person can leave at a different floor is:
Total number of ways = C(7, 5) × 5! = 21 × 120 = 2520
Thus, the total number of ways in which each of the five persons can leave at different floors is 2,520.
Final Answer:
(i) The total number of ways in which each of the five persons can leave at any one of the 7 floors is 16,807.
(ii) The total number of ways in which each of the five persons can leave at different floors is 2,520.
