Number of ways in which 12 different things can be distributed in 3 groups, is

Options

A. `(12!)/((4!)^(3))`
B. `(12!)/(3!(4!)^(3))`
C. `(12!)/(4!(3!)^(2))`
D. `(12!)/((3!)^(4))`

Answer

To solve the problem of distributing 12 different things into 3 groups, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Distribution**:
We want to distribute 12 different items into 3 groups. Each group must contain exactly 4 items since 12 divided by 3 equals 4.

2. **Selecting Items for the First Group**:
We first select 4 items from the 12 available items for the first group. The number of ways to choose 4 items from 12 is given by the combination formula:
\[ \text{Number of ways} = \binom{12}{4} = \frac{12!}{4! \cdot (12-4)!} = \frac{12!}{4! \cdot 8!} \]

3. **Selecting Items for the Second Group**:
After selecting the first group, we have 8 items left. Now, we select 4 items from these 8 for the second group:
\[ \text{Number of ways} = \binom{8}{4} = \frac{8!}{4! \cdot (8-4)!} = \frac{8!}{4! \cdot 4!} \]

4. **Selecting Items for the Third Group**:
Finally, we select the remaining 4 items for the third group. Since there are exactly 4 items left, there is only one way to choose them:
\[ \text{Number of ways} = \binom{4}{4} = 1 \]

5. **Calculating Total Ways**:
The total number of ways to distribute the items into the three groups is the product of the number of ways to select each group:
\[ \text{Total ways} = \binom{12}{4} \cdot \binom{8}{4} \cdot \binom{4}{4} \]
Substituting the values we calculated:
\[ \text{Total ways} = \frac{12!}{4! \cdot 8!} \cdot \frac{8!}{4! \cdot 4!} \cdot 1 \]

6. **Simplifying the Expression**:
Notice that the \(8!\) in the numerator and denominator cancels out:
\[ \text{Total ways} = \frac{12!}{4! \cdot 4! \cdot 4!} \]

7. **Final Result**:
Thus, the total number of ways to distribute 12 different items into 3 groups of 4 items each is:
\[ \text{Total ways} = \frac{12!}{(4!)^3} \]

### Conclusion:
The answer to the question is:
\[ \frac{12!}{(4!)^3} \]