Recursion - subsets - C++

[benfung]

I'm writing a simple program to calculate all possible combinations in a set by splitting the data into 2 sets (C(n-1, k-1) & C(n-1, k)) and calling the function recursively. Below is what I've written:

void RecSubsets(string soFar, string rest) {
if (rest == "") {
cout << soFar << end;
}
else {
for (int i = 0; i < rest.length(); i++) {
string next = soFar + rest[i];
string remaining = rest.substr(i+1);
RecSubsets(next, remaining);
}
}
}

void CanMakeSum(string set) {
RecSubsets("", set);
RecSubsets("", set.substr(1));
}

int main() {
string set = "abcd";
CanMakeSum(set);
return 0;
}
So for an input of 'abcd', it'd split the string into 2 groups (abcd, abc) and then print out the possible combinations by calling the function recursively. Using this algorithm, the output ought to be: abcd, abc, abd, ab, acd, ac, ad, a...
However, using the program above, the output is abcd, abd, acd, ad... I understand conceptually what I'm trying to achieve, but am having difficulty translating it into codes. Can someone pls point out where I'm going wrong? Thanks

[ResidentBiscuit]

There's an algorithm out there that can be used to create all subsets of any set actually. I think it's called the next subset algorithm. CS people can be pretty creative with their names.