How to get the lexical rank of a string?

How to get the rank of a string which is the position that the word occupies in the lexical ordering of all permutations of the characters in that word?

For example:

rank("bac") = 2, bascules the lexical ordering is abc, acb, bac, bca, cab, cba(we use zero base index here)

I wrote a C++ solution works for strings that have distinct characters like the above example.

    int getrank(string word) {
        int size = word.size();
        string temp = word;
        sort(temp.begin(), temp.end());
        int frac = 1;
        for (int i = 2; i <= word.size() ;i++) {
            frac *= i;
        }
        int res = 0;
        for (int i = size; i > 1; i--) {
            frac /= i;
            int choosed = 0;
            char check = word[size - i];
            while (temp[choosed] != check)
                choosed++;
            res += choosed * frac;
            temp.erase(choosed, 1);
        }
        return res;
    }

However I am struggling to compute the rank of strings contain duplicate chars.

For example:

rank("abba") should equal to 2 because only aabb, abab are in front of it.

How to solve this problem?