In how many ways can 16 books on different subjects be divided equally into 4 groups?

Options

A. `(16!)/((4!))`
B. `4! Xx (16 !)`
C. `(16!)/( (4!)^5)`
D. none of these

Answer

To solve the problem of dividing 16 books on different subjects into 4 equal groups, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to divide 16 distinct books into 4 groups, with each group containing 4 books.

2. **Identify the Total Number of Books and Groups**:
- Total books (M) = 16
- Total groups (N) = 4
- Books per group = 16 / 4 = 4

3. **Use the Formula for Division into Groups**:
The formula to divide M distinct items into N groups of equal size is given by:
\[ \frac{M!}{(k!)^N \cdot N!} \]
where \( k \) is the number of items in each group, \( M \) is the total number of items, and \( N \) is the number of groups.

4. **Substituting the Values**:
Here, \( M = 16 \), \( k = 4 \), and \( N = 4 \).
- So, we substitute into the formula:
\[ \frac{16!}{(4!)^4 \cdot 4!} \]

5. **Calculate Factorials**:
- First, calculate \( 4! \):
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]
- Now, \( (4!)^4 = 24^4 \).

6. **Final Calculation**:
- Now we can express the total number of ways to divide the books:
\[ \text{Number of ways} = \frac{16!}{(24)^4 \cdot 24} \]
- This simplifies to:
\[ \frac{16!}{24^5} \]

7. **Conclusion**:
The final answer for the number of ways to divide 16 books into 4 equal groups is:
\[ \frac{16!}{(4!)^4 \cdot 4!} \]