The number of ways in which 12 different objects can be divided into three groups each containing 4 objects is

Options

A. `((12!))/((4!)^(3)(!3))`
B. `((12!))/((4!)^(3))`
C. `((12!))/((4!))`
D. none of the above

Answer

To solve the problem of dividing 12 different objects into three groups, each containing 4 objects, we can follow these steps:

### Step-by-Step Solution:

1. **Choosing the First Group**:
We start by selecting 4 objects from the 12 available objects. The number of ways to choose 4 objects from 12 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Ways to choose the first group} = \binom{12}{4} \]

2. **Choosing the Second Group**:
After selecting the first group, we have 8 objects left. Now, we need to choose 4 objects from these 8. The number of ways to do this is:
\[ \text{Ways to choose the second group} = \binom{8}{4} \]

3. **Choosing the Third Group**:
After selecting the first and second groups, there are 4 objects remaining. We choose all 4 of these objects for the third group. The number of ways to choose the third group is:
\[ \text{Ways to choose the third group} = \binom{4}{4} = 1 \]

4. **Calculating Total Combinations**:
Now, we multiply the number of ways to choose each group:
\[ \text{Total ways} = \binom{12}{4} \times \binom{8}{4} \times \binom{4}{4} \]

5. **Adjusting for Group Order**:
Since the order of the groups does not matter (i.e., Group 1, Group 2, and Group 3 are indistinguishable), we need to divide by the number of ways to arrange the 3 groups, which is \( 3! \):
\[ \text{Final Count} = \frac{\binom{12}{4} \times \binom{8}{4} \times \binom{4}{4}}{3!} \]

6. **Substituting the Combination Formula**:
We can use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \):
\[ \binom{12}{4} = \frac{12!}{4! \times 8!}, \quad \binom{8}{4} = \frac{8!}{4! \times 4!}, \quad \binom{4}{4} = 1 \]

7. **Putting it All Together**:
Substitute these values into the total ways equation:
\[ \text{Total ways} = \frac{\frac{12!}{4! \times 8!} \times \frac{8!}{4! \times 4!} \times 1}{3!} \]
Simplifying this gives:
\[ = \frac{12!}{(4!)^3 \times 3!} \]

### Final Answer:
Thus, the number of ways to divide 12 different objects into three groups of 4 is:
\[ \frac{12!}{(4!)^3 \times 3!} \]