In how any different ways can a set `A` of `3n` elements be partitioned into 3 subsets of equal number of elements? The subsets `P ,Q ,R` form a partition if `PuuQuuR=A ,PnnR=varphi,QnnR=varphi,RnnP=varphidot`

Explanation

To solve the problem of partitioning a set \( A \) of \( 3n \) elements into 3 subsets \( P, Q, R \) of equal size, we can follow these steps:

### Step 1: Understand the problem
We need to partition \( 3n \) elements into 3 subsets, each containing \( n \) elements. This means that each subset will have the same number of elements, and together they will cover all elements in set \( A \).

**Hint:** Remember that the total number of elements must be evenly divisible by the number of subsets.

### Step 2: Calculate the total arrangements
First, we can arrange all \( 3n \) elements in a line. The total number of arrangements of \( 3n \) elements is given by \( (3n)! \).

**Hint:** Think about how many ways you can arrange a set of distinct items.

### Step 3: Account for identical subsets
Since the subsets \( P, Q, R \) are indistinguishable from each other, we need to divide by the number of ways to arrange these subsets. There are \( 3! \) (which equals 6) ways to arrange 3 subsets.

**Hint:** When dealing with identical groups, always remember to divide by the factorial of the number of groups.

### Step 4: Calculate arrangements within each subset
Next, we need to account for the arrangements within each subset. Each subset contains \( n \) elements, and the number of ways to arrange \( n \) elements in a subset is \( n! \). Since there are 3 subsets, we need to account for this by dividing by \( (n!)^3 \).

**Hint:** Consider how many ways you can arrange items within a group, and remember to do this for each group.

### Step 5: Combine the calculations
Putting all of this together, the total number of ways to partition the set \( A \) into the subsets \( P, Q, R \) is given by the formula:

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

### Final Answer
Thus, the number of different ways to partition a set \( A \) of \( 3n \) elements into 3 subsets of equal number of elements is:

\[ \frac{(3n)!}{(n!)^3 \cdot 6} \]

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