There are 3 apartments A, B and C for rent in a building . Each apartment will accept either 3 or 4 occupants. The number of ways of renting the apartments to 10 students

Options

A. 12600
B. 10800
C. 13500
D. 15000

Answer

To solve the problem of renting 3 apartments (A, B, and C) to 10 students, where each apartment can accommodate either 3 or 4 occupants, we need to determine the different ways to distribute the students among the apartments.

### Step-by-Step Solution:

1. **Identify the possible distributions of students**:
Since we have 10 students and each apartment can take either 3 or 4 students, the only possible distribution that satisfies the total of 10 students is:
- Apartment A: 3 students
- Apartment B: 3 students
- Apartment C: 4 students

2. **Choose the group of 4 students**:
We first need to choose which 4 students will occupy Apartment C. The number of ways to choose 4 students from 10 is given by the combination formula:
\[ \text{Number of ways} = \binom{10}{4} \]
Calculating this:
\[ \binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]

3. **Distribute the remaining 6 students**:
After selecting 4 students for Apartment C, we have 6 students left. We need to choose 3 of these 6 students to occupy Apartment A. The number of ways to choose 3 students from 6 is:
\[ \text{Number of ways} = \binom{6}{3} \]
Calculating this:
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

4. **Assign students to apartments**:
Now, we have assigned 4 students to Apartment C and 3 students to Apartment A. The remaining 3 students will automatically go to Apartment B.

5. **Calculate the total arrangements**:
Since the apartments are distinct, we can arrange the students in the apartments in different ways. The arrangement of students in each apartment can be calculated as follows:
- For Apartment C (4 students): \(4!\) ways
- For Apartment A (3 students): \(3!\) ways
- For Apartment B (3 students): \(3!\) ways

Therefore, the total arrangements are:
\[ \text{Total arrangements} = 4! \times 3! \times 3! = 24 \times 6 \times 6 = 864 \]

6. **Combine the selections and arrangements**:
The total number of ways to rent the apartments is the product of the ways to choose the students and the arrangements:
\[ \text{Total ways} = \binom{10}{4} \times \binom{6}{3} \times (4! \times 3! \times 3!) \]
Substituting the values:
\[ \text{Total ways} = 210 \times 20 \times 864 \]

7. **Calculate the final result**:
Now we calculate:
\[ 210 \times 20 = 4200 \]
Then,
\[ 4200 \times 864 = 3628800 \]

Thus, the total number of ways to rent the apartments to 10 students is **3628800**.