Finding all the 3-combinations of the output of for loops

If value of k is varying and much larger than 3, we'll have to apply recursion. Till then 3 for loops works fine.

#include <iostream>
#include <vector>

using namespace std;

int main()
{
    vector<vector<int>> M;
    for (int k = 0; k < 2; ++k)
        for (int a = 0; a < 4; ++a)
            for (int b = 0; b < 4; ++b)
                if (a != b)
                    M.push_back(vector<int>({ k, a, b }));

    vector<vector<int>> res;
    for (int i = 0; i < (int)M.size() - 2; ++i)
        for (int j = i + 1; j < (int)M.size() - 1; ++j)
            for (int k = j + 1; k < (int)M.size(); ++k)
            {
                vector<int> temp;

                for (int e : M[i])
                    temp.push_back(e);
                for (int e : M[j])
                    temp.push_back(e);
                for (int e : M[k])
                    temp.push_back(e);
                res.push_back(temp);
            }

    for (const vector<int>& V : res)
    {
        for (int e : V)
            cout << e << ' ';
        cout << endl;
    }
}

To make things easier, I'd group a k, a and b value together into a single object. Let's call it a K-A-B object, or a kab:

struct kab {
    int k, a, b;
};

Then, I'd make an increment function that'll take a kab and give us the next kab in the sequence. This'll allow us to condense all three of your loops down into one and make it easier to iterate from any point in the sequence. That could look like:

kab increment(kab obj) {
    obj.b = (obj.b + 1) % (MAX_B + 1);
    if(obj.b == 0) {
        obj.a = (obj.a + 1) % (MAX_A + 1);
        if(obj.a == 0) {
            obj.k++;
        }
    }
 
    // Ensure a != b by incrementing again if they are
    if(obj.a == obj.b) {
        return increment(obj);
    }
 
    return obj;
}

Lastly, it's just a matter of doing the actual increments. Here's an example:

while(vals.k <= MAX_K) {
    kab vals2 = increment(vals);
    while(vals2.k <= MAX_K) {
        kab vals3 = increment(vals2);
        while(vals3.k <= MAX_K) {
            std::cout << "k: " << vals.k  << " a: " << vals.a  << " b: " << vals.b  << std::endl;
            std::cout << "k: " << vals2.k  << " a: " << vals2.a  << " b: " << vals2.b  << std::endl;
            std::cout << "k: " << vals3.k  << " a: " << vals3.a  << " b: " << vals3.b  << std::endl;
            std::cout << std::endl;

            vals3 = increment(vals3);
        }
        vals2 = increment(vals2);
    }
    vals = increment(vals);
}

See it run here: https://ideone.com/9cVHuG (note that that much output causes ideone to timeout. This should work fine locally).