In how many ways can 8 books be divided between 2 students, if they get 3 and 5 books.

Answer

To solve the problem of how many ways 8 books can be divided between 2 students, where one student gets 3 books and the other gets 5 books, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Books**:
We have a total of 8 books.

2. **Determine the Distribution**:
We need to divide these books between two students: Student 1 (S1) will receive 3 books, and Student 2 (S2) will receive 5 books.

3. **Choose Books for Student 1**:
We need to select 3 books out of the 8 for Student 1. The number of ways to choose 3 books from 8 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 8 \) and \( r = 3 \):
\[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \cdot 5!} \]

4. **Calculate the Factorial Values**:
We can simplify this:
\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]

5. **Assign Remaining Books to Student 2**:
After selecting 3 books for Student 1, the remaining 5 books automatically go to Student 2. There is only 1 way to assign these remaining books.

6. **Final Calculation**:
Therefore, the total number of ways to divide the books is simply the number of ways to choose the 3 books for Student 1, which we calculated as 56.

### Conclusion:
The total number of ways to divide the 8 books between the two students, where one gets 3 books and the other gets 5 books, is **56**.

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