In how many ways can 12 books be divided in 3 students equally?

Answer

To solve the problem of dividing 12 books among 3 students equally, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the number of books per student**: Since we have 12 books and 3 students, each student will receive:
\[ \text{Books per student} = \frac{12}{3} = 4 \]

2. **Choose books for the first student**: We need to select 4 books out of the 12 for the first student. The number of ways to choose 4 books from 12 is given by the combination formula:
\[ \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \cdot 8!} \]

3. **Choose books for the second student**: After assigning 4 books to the first student, we have 8 books left. Now, we need to choose 4 books for the second student from these 8:
\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4! \cdot 4!} \]

4. **Assign books to the third student**: The remaining 4 books will automatically go to the third student. There is only 1 way to do this since all remaining books must be assigned:
\[ \binom{4}{4} = 1 \]

5. **Calculate the total arrangements**: The total number of ways to distribute the books is the product of the combinations calculated in the previous steps. However, since the students are indistinguishable in terms of the groups formed, we need to divide by the number of ways to arrange the 3 groups (students):
\[ \text{Total ways} = \frac{\binom{12}{4} \cdot \binom{8}{4} \cdot \binom{4}{4}}{3!} \]

6. **Substituting values**:
\[ \text{Total ways} = \frac{\frac{12!}{4! \cdot 8!} \cdot \frac{8!}{4! \cdot 4!} \cdot 1}{3!} \]
Simplifying this gives:
\[ = \frac{12!}{(4!)^3 \cdot 3!} \]

7. **Final calculation**: Now we can compute the values:
- \( 12! = 479001600 \)
- \( 4! = 24 \)
- \( 3! = 6 \)

Therefore,
\[ \text{Total ways} = \frac{479001600}{24^3 \cdot 6} = \frac{479001600}{13824 \cdot 6} = \frac{479001600}{82944} = 5760 \]

### Final Answer:
The total number of ways to divide 12 books among 3 students equally is **5760**.