In how many ways 12 different books can be distributed equally among 3 persons?

Options

A. `(12!)/((4!)^(3))`
B. `(12!)/((3!)^(4))`
C. `(12!)/((4!)^(4))`
D. `(12!)/((3!)^(3))`

Answer

To solve the problem of distributing 12 different books equally among 3 persons, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Distribution**: We need to distribute 12 different books among 3 persons such that each person gets 4 books.

2. **Select Books for the First Person**:
- The number of ways to select 4 books out of 12 for the first person can be calculated using the combination formula \( C(n, r) = \frac{n!}{r!(n-r)!} \).
- Here, \( n = 12 \) and \( r = 4 \).
- So, the number of ways to choose 4 books from 12 is:
\[ C(12, 4) = \frac{12!}{4! \cdot (12-4)!} = \frac{12!}{4! \cdot 8!} \]

3. **Select Books for the Second Person**:
- After selecting 4 books for the first person, we have 8 books left.
- The number of ways to select 4 books from these 8 for the second person is:
\[ C(8, 4) = \frac{8!}{4! \cdot (8-4)!} = \frac{8!}{4! \cdot 4!} \]

4. **Select Books for the Third Person**:
- Finally, the last 4 books will automatically go to the third person. The number of ways to select 4 books from 4 is:
\[ C(4, 4) = \frac{4!}{4! \cdot 0!} = 1 \]

5. **Combine the Selections**:
- The total number of ways to distribute the books is the product of the ways to select books for each person:
\[ \text{Total Ways} = C(12, 4) \cdot C(8, 4) \cdot C(4, 4) \]
- Substituting the values we calculated:
\[ \text{Total Ways} = \frac{12!}{4! \cdot 8!} \cdot \frac{8!}{4! \cdot 4!} \cdot 1 \]

6. **Simplify the Expression**:
- Notice that \( 8! \) cancels out:
\[ \text{Total Ways} = \frac{12!}{4! \cdot 4! \cdot 4!} \]

7. **Account for the Order of Persons**:
- Since the three persons are distinct, we must multiply by the number of ways to arrange the 3 persons, which is \( 3! \):
\[ \text{Final Total Ways} = 3! \cdot \frac{12!}{4! \cdot 4! \cdot 4!} \]
- This gives us:
\[ \text{Final Total Ways} = 6 \cdot \frac{12!}{4! \cdot 4! \cdot 4!} \]

### Final Answer:
The total number of ways to distribute 12 different books equally among 3 persons is:
\[ 6 \cdot \frac{12!}{4! \cdot 4! \cdot 4!} \]