In a 12 -storey house ten people enter a lift cabin. It is known that they will left in groups of 2,3 and 5 people at different storeys. The number of ways they can do so if the left does not stop to the second storey is

Options

A. A. 170
B. B. 22
C. C. 720
D. D. none of these

Answer

To solve the problem of how many ways 10 people can exit a lift in groups of 2, 3, and 5 at different storeys of a 12-storey building (excluding the second storey), we can follow these steps:

### Step 1: Identify the Total Number of Storeys Available
The total number of storeys in the building is 12. However, since the lift does not stop at the second storey, we have:
- Total storeys = 12
- Excluded storey = 1 (the second storey)

Thus, the available storeys for the lift to stop at are:
- Available storeys = 12 - 1 = 11 storeys (1st, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th, 11th, and 12th)

### Step 2: Determine the Groups
The problem states that the 10 people will exit in groups of:
- 2 people
- 3 people
- 5 people

### Step 3: Calculate the Number of Ways to Arrange the Groups
We need to find the number of ways to arrange these groups of people across the available storeys. Since the groups are fixed in size (2, 3, and 5), we can treat them as distinct groups.

1. **Total number of groups**:
- Group of 2: 1 group
- Group of 3: 1 group
- Group of 5: 1 group

2. **Total ways to arrange the groups**:
- The number of ways to arrange the groups of 2, 3, and 5 people in 11 available storeys can be calculated using the formula for permutations. Since we have 3 groups, we can choose 3 storeys from the 11 available storeys.

### Step 4: Calculate the Permutations
To find the number of ways to choose 3 storeys from 11, we can use the permutation formula:
\[ P(n, r) = \frac{n!}{(n-r)!} \]
Where \(n\) is the total number of storeys (11) and \(r\) is the number of groups (3).

So,
\[ P(11, 3) = \frac{11!}{(11-3)!} = \frac{11!}{8!} = 11 \times 10 \times 9 = 990 \]

### Step 5: Final Answer
Thus, the total number of ways the 10 people can exit the lift in groups of 2, 3, and 5 at different storeys (excluding the second storey) is **990**.