Creating permutations for the Contdown problem

I'm trying to make a small program to solve the countdown problem. Basically, given a number n and a vector of numbers, the problem outputs if n can be constructed applying simple operations (say addition, div, mult, and subs).

The input I'm trying to give to the program is a string with reverse polish notation, so I made a small function to solve this kind of expressions.

Here is my code:

#include <iostream>
#include <stack>
#include <string>
#include <vector>
using namespace std;

bool is_number(const std::string& s) {
    string::const_iterator it = s.begin();
    while (it != s.end() && std::isdigit(*it)) ++it;
    return !s.empty() && it == s.end();
}

vector<string> parseInput(const string& expression) {
    vector<string> parsedInp;
    string current = "";
    for (auto token : expression) {
        if (isdigit(token)) {
            current += token;
        }
        else if (token == ' ') {
            if (current != "") {
                parsedInp.push_back(current);
                current = "";
            }
        }
        else {
            parsedInp.push_back(string(1, token));
        }
    }
    return parsedInp;
}

double operateExpression(const vector<string>& parsedInp) {
    stack<double> expression;
    for (auto token : parsedInp) {
        if (is_number(token)) {
            expression.push(stod(token));
        }
        else {
            double op1 = expression.top();
            expression.pop();
            double op2 = expression.top();
            expression.pop();
            if (token == "+") {
                expression.push(op1 + op2);
            }           
            if (token == "-") {
                expression.push(op1 - op2);
            }           
            if (token == "*") {
                expression.push(op1 * op2);
            }           
            if (token == "/") {
                expression.push(op1 / op2);
            }
        }
    }
    return expression.top();
}


vector<int> getSolution(vector<string> inp, const int target) {
    vector<string> op = { "+", "-", "*", "/", "" };
    //TODO....
    
}


int main() {
    string expression;
    getline(cin, expression);
    vector<string> parsedInp = parseInput(expression);
    cout << operateExpression(parsedInp) << endl;
}

So my first instinct was to create all possible permutations of the vector, and then try to concatenate the operations in the vector, but the problem that I'm facing now is that I can not find an easy way to do it. I can't concatenate it carelessly, because operateExpression would fail, but I don't want to use try and catch for this problem. Also, I think there should be an easier way to do it.

So is there any better way to write getSolution here? Thanks.