In the Suniti building in Mumbai there are 12 floors plus the ground floor. 9 people get into the lift of the building on the ground floor. The lift does not stop on the first floor. If 2, 3 and 4 people alight from the lift on its upward journey, then in how many ways can they do so? (Assume they alight on different floors.)

Options

A. `""^(11)C_(3) xx ""^(3) P_(3)`
B. ` ""^(11)P_(3) xx ""^(9) C_(4) xx""^(5)C_(3)`
C. `""^(10)P_(3)xx""^(9)C_(4)xx ""^(5)C_(3)`
D. `""^(12)C_(3)`

Answer

To solve the problem, we need to determine the number of ways in which 9 people can alight from a lift on different floors, given that the lift does not stop on the first floor and that 2, 3, and 4 people alight on different floors during its journey.

### Step-by-Step Solution:

1. **Identify the Total Number of Floors:**
The building has 12 floors plus the ground floor. Since the lift does not stop at the first floor, we have:
\[ \text{Total floors available} = 12 - 1 = 11 \text{ floors} \]

2. **Determine the Number of People Alighting:**
We have 9 people getting into the lift, and they will alight in groups of 2, 3, and 4. This means:
- 2 people will alight on one floor,
- 3 people will alight on another floor,
- 4 people will alight on yet another floor.

3. **Choose the Floors for Alighting:**
We need to choose 3 different floors from the available 11 floors for the 2, 3, and 4 people to alight. The number of ways to choose 3 floors from 11 is given by the permutation formula:
\[ \text{Ways to choose floors} = P(11, 3) = 11P3 \]

4. **Choose the People for Each Group:**
- **Choosing 4 People:** We first choose 4 people from the 9 to alight on the first chosen floor. The number of ways to choose 4 people from 9 is:
\[ \text{Ways to choose 4 people} = C(9, 4) \]

- **Choosing 3 People:** After 4 people have alighted, 5 people remain. We then choose 3 people from these 5 to alight on the second chosen floor. The number of ways to choose 3 people from 5 is:
\[ \text{Ways to choose 3 people} = C(5, 3) \]

- **Choosing 2 People:** The remaining 2 people will alight on the last chosen floor. There is only one way for this since they are the only ones left.

5. **Combine All the Choices:**
The total number of ways for the entire process is the product of the ways to choose the floors and the ways to choose the people:
\[ \text{Total ways} = P(11, 3) \times C(9, 4) \times C(5, 3) \]

### Final Calculation:
Putting it all together, we have:
\[ \text{Total ways} = 11P3 \times 9C4 \times 5C3 \]

### Conclusion:
Thus, the total number of ways in which the 9 people can alight from the lift on different floors is given by the expression above.

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