S is a set of n elements consisting of integers from 1 to n. In m subsets selected from this set with 6 elements and each element being different, the sum of the elements of the subset is divided by 3.

What is the equivalent of the value remaining when the sum of the elements of the set satisfying this condition is divided by 9 in terms of m and n?

Given the problem, we need to find the remainder when the sum of elements of subsets, which are divisible by 3, is divided by 9, in terms of \( m \) (the number of such subsets) and \( n \) (the number of elements in the set \( S \)).

### Problem Breakdown:

1. **Set \( S \):** This set contains \( n \) elements from \( 1 \) to \( n \).
2. **Subsets:** We consider subsets of \( S \) with exactly 6 elements, where each subset’s sum of elements is divisible by 3.
3. **Objective:** We need to determine the remainder when the sum of all elements of these subsets is divided by 9, expressed in terms of \( m \) and \( n \).

### Analysis:

The subsets are formed by picking 6 elements from the set \( S = \{1, 2, \dots, n\} \).

#### Step 1: Properties of the set \( S \)
Each element \( x \) in \( S \) has a residue when divided by 3. The residues can be 0, 1, or 2. The distribution of residues is important since the sum of elements in any subset must be divisible by 3.

#### Step 2: Sum of Elements in Subsets
If we pick a subset \( T \) of 6 elements from \( S \), the sum \( \text{Sum}(T) \) must satisfy:
\[
\text{Sum}(T) \equiv 0 \ (\text{mod}\ 3)
\]
The problem asks about the remainder of the sum of elements of these subsets modulo 9.

#### Step 3: Counting the Remainders Modulo 9
Since the sum of each subset is divisible by 3, the sum of all such sums modulo 9 will be considered. Let the sum of all elements in such \( m \) subsets be \( S_{\text{total}} \). We are interested in \( S_{\text{total}} \mod 9 \).

### Result:
If the sum of elements in each of these subsets is \( S_{\text{set}} \), the total sum can be written as:
\[
S_{\text{total}} = m \times S_{\text{set}}
\]
Given that \( S_{\text{set}} \) is divisible by 3, we know:
\[
S_{\text{total}} = 3k \quad \text{for some integer } k
\]

Thus, the remainder when divided by 9 is given by:
\[
S_{\text{total}} \mod 9 = 3k \mod 9
\]
The remainder when this total sum \( S_{\text{total}} \) is divided by 9 is:
\[
S_{\text{total}} \mod 9 = 3 \times (k \mod 3)
\]

### Conclusion:
The remainder when the sum of the elements of these subsets is divided by 9 is equivalent to:
\[
3 \times (m \mod 3)
\]

Thus, the required remainder is \( 3 \times (m \mod 3) \).

---

Would you like any further details on this? 

Here are some additional questions that might interest you:

1. How do different values of \( m \) affect the remainder when divided by 9?
2. What happens if we change the subset size from 6 to another value?
3. How do the properties of \( n \) influence the distribution of remainders?
4. What are the implications of a non-divisible sum for the set modulo 9?
5. How does this result generalize for other moduli like 12 or 15?

**Tip:** Understanding modular arithmetic can significantly simplify solving problems involving sums and remainders.

Understand how to find the remainder modulo 9 when the sum of elements in subsets of a set S, each of 6 elements, is divisible by 3. Explore the application of modular arithmetic and combinatorics to solve this advanced problem.

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