The number of ways to distribute m `xx` n different thing among n persons equally =`((mn)!)/((m!)^(n))`
The number of ways in which 12 different books can be divided equally in 3 heaps of book is

Options

A. `(3.12!)/((4!)^(3))`
B. `(12!)/(3!(4!)^(3))`
C. `(12!)/(3*(4!)^(3))`
D. `(3!*12!)/((4!)^(3))`

Answer

To solve the problem of distributing 12 different books into 3 equal heaps, we can use the formula for distributing m different things among n persons equally. The formula is given by:

\[ \frac{(mn)!}{(m!)^n} \]

where \( m \) is the number of items in each heap, and \( n \) is the number of heaps.

### Step 1: Determine the values of m and n
In this case, we have:
- Total number of books = 12
- Number of heaps (n) = 3

Since we need to divide 12 books equally into 3 heaps, each heap will contain:

\[ m = \frac{12}{3} = 4 \]

So, we have \( m = 4 \) and \( n = 3 \).

### Step 2: Apply the formula
Now we can substitute the values of \( m \) and \( n \) into the formula:

\[ \text{Number of ways} = \frac{(mn)!}{(m!)^n} = \frac{(4 \cdot 3)!}{(4!)^3} \]

### Step 3: Calculate the factorials
Now, we need to calculate the factorials:

- \( mn = 4 \cdot 3 = 12 \)
- \( 12! = 479001600 \)
- \( 4! = 24 \)
- \( (4!)^3 = 24^3 = 13824 \)

### Step 4: Substitute the factorials into the formula
Now we can substitute these values back into our equation:

\[ \text{Number of ways} = \frac{12!}{(4!)^3} = \frac{479001600}{13824} \]

### Step 5: Perform the division
Now we perform the division:

\[ \frac{479001600}{13824} = 34650 \]

### Conclusion
Thus, the number of ways to divide 12 different books into 3 heaps equally is:

\[ \text{34650} \]