The number of ways of distributing 12 different objects among three person such that one particular person is always getting 6 objects and the remaining two persons are getting three objects each is

Options

A. `(12!)/(6!(3!)^(2)2!)`
B. `(12!)/(6!(3!)^(2))`
C. `(12!)/((6!)^(2))`
D. `((12)!)/((3!)^(2)(6!)^(2))`

Answer

To solve the problem of distributing 12 different objects among three persons such that one particular person always gets 6 objects and the remaining two persons get 3 objects each, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Persons and Objects**:
Let the three persons be A, B, and C. We want to give 6 objects to person A, and 3 objects each to persons B and C.

2. **Select Objects for Person A**:
We need to choose 6 objects from the 12 available objects to give to person A. The number of ways to choose 6 objects from 12 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of objects, and \( r \) is the number of objects to choose.
\[ \text{Ways for A} = \binom{12}{6} \]

3. **Select Objects for Person B**:
After giving 6 objects to A, there are 6 objects left. Now we need to choose 3 objects from these 6 to give to person B. The number of ways to choose 3 objects from 6 is:
\[ \text{Ways for B} = \binom{6}{3} \]

4. **Assign Remaining Objects to Person C**:
After assigning 3 objects to B, there are 3 objects left, which will automatically go to person C. There is only one way to give these remaining 3 objects to C:
\[ \text{Ways for C} = 1 \]

5. **Calculate the Total Number of Ways**:
The total number of ways to distribute the objects is the product of the ways to distribute to each person:
\[ \text{Total Ways} = \binom{12}{6} \times \binom{6}{3} \times 1 \]

6. **Compute the Combinations**:
- Calculate \( \binom{12}{6} = \frac{12!}{6!6!} \)
- Calculate \( \binom{6}{3} = \frac{6!}{3!3!} \)

Thus, we have:
\[ \text{Total Ways} = \frac{12!}{6!6!} \times \frac{6!}{3!3!} \times 1 = \frac{12!}{3!3!6!} \]

7. **Final Calculation**:
Now, we can simplify this expression to get the final answer.

### Final Answer
The total number of ways to distribute the 12 different objects among the three persons under the given conditions is:
\[ \frac{12!}{3!3!6!} \]