If 3n different things can be equally distributed among 3 persons in k ways, then the number of ways to divide the 3n things in 3 equal groups is:

Options

A. `kxx3!`
B. `(k)/(3!)`
C. `(3!)k`
D. `3k`

Answer

To solve the problem, we need to find the number of ways to divide \(3n\) different things into 3 equal groups, given that these things can be distributed among 3 persons in \(k\) ways.

### Step-by-Step Solution:

1. **Understanding the Problem**: We have \(3n\) different items that need to be divided into 3 equal groups. Each group will contain \(n\) items.

2. **Distribution Among Persons**: The problem states that these \(3n\) items can be distributed among 3 persons in \(k\) ways. This means that if we consider each person as a group, the distribution can be done in \(k\) ways.

3. **Dividing into Groups**: We want to find the number of ways to divide these \(3n\) items into 3 equal groups. Let's denote the number of ways to divide the items into 3 equal groups as \(x\).

4. **Using the Formula for Distribution**: When dividing \(3n\) items into 3 groups, the first group can be chosen in \(\binom{3n}{n}\) ways. After choosing the first group of \(n\) items, we have \(2n\) items left. The second group can then be chosen in \(\binom{2n}{n}\) ways. The last group will automatically consist of the remaining \(n\) items.

5. **Accounting for Indistinguishable Groups**: Since the groups are indistinguishable, we need to divide by the number of ways to arrange the 3 groups, which is \(3!\) (factorial of 3). Therefore, the number of ways to divide \(3n\) items into 3 equal groups is given by:
\[ x = \frac{\binom{3n}{n} \cdot \binom{2n}{n}}{3!} \]

6. **Relating to Given Information**: From the problem, we know that the distribution of \(3n\) items among 3 persons is \(k\). The relationship can be established as follows:
\[ k = 3! \cdot x \]
This means that \(x\) can be expressed in terms of \(k\):
\[ x = \frac{k}{3!} \]

7. **Final Expression**: We can substitute \(3!\) with \(6\):
\[ x = \frac{k}{6} \]

### Conclusion:
The number of ways to divide \(3n\) different things into 3 equal groups is:
\[ \boxed{\frac{k}{6}} \]