Question Analysis:
We are given that 3n different things can be equally distributed among 3 persons in k ways. We need to find the number of ways to divide the 3n things into 3 equal groups.

Key Concepts:
- Distribution of objects: There are n objects and r persons, and we need to distribute the objects among the persons.
- Combination: The number of ways to choose r objects from a set of n objects is denoted by nCr or C(n, r).
- Permutation: The number of ways to arrange n objects taken r at a time is denoted by nPr or P(n, r).

Solution:
To find the number of ways to divide the 3n things into 3 equal groups, we can use the concept of distributing objects among persons.

Step 1: Distributing the 3n things among 3 persons
We are given that 3n different things can be equally distributed among 3 persons in k ways. This means that each person will receive n objects.

The number of ways to distribute n objects among 3 persons can be found using combination. We need to choose n objects from the 3n objects, and distribute them among the 3 persons. Hence, the number of ways is given by:
C(3n, n)

Step 2: Permuting the 3 persons
Once the n objects are distributed among the 3 persons, we can permute the 3 persons in 3! ways.

Step 3: Multiply the number of ways
To find the total number of ways to divide the 3n things into 3 equal groups, we need to multiply the number of ways from Step 1 with the number of ways from Step 2:
Total number of ways = C(3n, n) * 3!

Step 4: Simplify the expression
Using the property of combination, we have:
C(3n, n) = (3n)! / [(n!)^2]

Substituting this value in the expression, we get:
Total number of ways = [(3n)! / [(n!)^2]] * 3!

Simplifying further, we get:
Total number of ways = (3n)! / n!

Step 5: Comparing with the given options
The expression for the total number of ways to divide the 3n things into 3 equal groups is (3n)! / n!.

Comparing this expression with the given options, we find that the correct answer is option B, k/3!.