The set `S:{1,2,3…………12}` is to be partitioned into three sets A,B,C of equal size. Thus `A cup B cup C=S, A cap B =B cap C=A cap C=phi` The number of ways to partition S is :

Options

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

Answer

To solve the problem of partitioning the set \( S = \{1, 2, 3, \ldots, 12\} \) into three sets \( A, B, C \) of equal size, we need to follow these steps:

### Step 1: Determine the size of each set
Since the total number of elements in set \( S \) is 12 and we want to partition it into three sets of equal size, each set will have:
\[ \text{Size of each set} = \frac{12}{3} = 4 \]

### Step 2: Choose elements for set A
We first choose 4 elements from the 12 elements to form set \( A \). The number of ways to choose 4 elements from 12 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Ways to choose A} = \binom{12}{4} \]

### Step 3: Choose elements for set B
After choosing set \( A \), we have 8 elements left. Now, we need to choose 4 elements from these 8 to form set \( B \):
\[ \text{Ways to choose B} = \binom{8}{4} \]

### Step 4: Assign remaining elements to set C
The remaining 4 elements will automatically go to set \( C \). Therefore, there is only 1 way to assign these elements:
\[ \text{Ways to choose C} = 1 \]

### Step 5: Account for the indistinguishability of sets
Since the sets \( A, B, C \) are indistinguishable (the order in which we choose them does not matter), we need to divide by the number of ways to arrange these 3 sets, which is \( 3! \):
\[ \text{Total arrangements} = \frac{\binom{12}{4} \cdot \binom{8}{4}}{3!} \]

### Step 6: Calculate the combinations
Now we can calculate the combinations:
\[ \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4!8!} = 495 \]
\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} = 70 \]

### Step 7: Substitute values into the total arrangements formula
Now we substitute the values into the total arrangements formula:
\[ \text{Total arrangements} = \frac{495 \cdot 70}{6} \]

### Step 8: Perform the final calculation
Calculating the above expression:
\[ 495 \cdot 70 = 34650 \]
\[ \frac{34650}{6} = 5775 \]

### Final Answer
The number of ways to partition the set \( S \) into three sets \( A, B, C \) of equal size is:
\[ \boxed{5775} \]