# Count of sub-sequences which satisfy the given condition

• Last Updated : 11 Jun, 2021

Given a string str consisting of digits, the task is to find the number of possible 4 digit sub-sequences which are of the form (x, x, x + 1, x + 1) where x can be from the range [0, 8]
Examples:

Input: str = “1122”
Output:
Only one sub-sequence is valid, i.e the entire string itself.
Input: str = “13134422”
Output:
Two Valid sub-sequences are present “1122” and “3344”.

Approach:

• We will find out total number of possible sub-sequences for each possible x from 0 to 8.
• For each x, remove all other digits from the String, except x and x+1 as they do not affect the answer.
• Maintain a prefix Sum array to count the number of x+1 digits till i th index in the String.
• Now, for every club of digits say size K (which are x), we can choose two numbers in KC2 ways. Last two numbers can be any two numbers from all the digits (which are x+1) which follows that club of digits (count is determined using Prefix Sum Array) say size L, so there are LC2 ways to choose. Total Ways = KC2 * LC2 .
• Till, Now we can be considered x to come from the same club, but it can also be from multiple Clubs. So, we have to consider all possible pairs of clubs and multiply their size to get number of ways to choose first two numbers. For last two numbers, ways will remain same.
• In order to prevent the problem of over counting in Step 5. Only Possible way which includes the current club under consideration will be chosen as other have already been considered in calculation of previous clubs.
• Add all the ways possible for all the values of x and take Modulo.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``#define ll long long int``#define MOD 1000000007``using` `namespace` `std;` `// Function to return the total``// required sub-sequences``int` `solve(string test)``{``    ``int` `size = test.size();``    ``int` `total = 0;` `    ``// Find ways for all values of x``    ``for` `(``int` `i = 0; i <= 8; i++) {``        ``int` `x = i;` `        ``// x+1``        ``int` `y = i + 1;``        ``string newtest;` `        ``// Removing all unnecessary digits``        ``for` `(``int` `j = 0; j < size; j++) {``            ``if` `(test[j] == x + 48 || test[j] == y + 48) {``                ``newtest += test[j];``            ``}``        ``}` `        ``if` `(newtest.size() > 0) {``            ``int` `size1 = newtest.size();` `            ``// Prefix Sum Array for X+1 digit``            ``int` `prefix[size1] = { 0 };``            ``for` `(``int` `j = 0; j < size1; j++) {``                ``if` `(newtest[j] == y + 48) {``                    ``prefix[j]++;``                ``}``            ``}` `            ``for` `(``int` `j = 1; j < size1; j++) {``                ``prefix[j] += prefix[j - 1];``            ``}` `            ``int` `count = 0;``            ``int` `firstcount = 0;` `            ``// Sum of squares``            ``int` `ss = 0;` `            ``// Previous sum of all possible pairs``            ``int` `prev = 0;` `            ``for` `(``int` `j = 0; j < size1; j++) {``                ``if` `(newtest[j] == x + 48) {``                    ``count++;``                    ``firstcount++;``                ``}``                ``else` `{` `                    ``ss += count * count;` `                    ``// To find sum of multiplication of all``                    ``// possible pairs``                    ``int` `pairsum = (firstcount * firstcount - ss) / 2;``                    ``int` `temp = pairsum;` `                    ``// To prevent overcounting``                    ``pairsum -= prev;``                    ``prev = temp;` `                    ``int` `secondway = prefix[size1 - 1];``                    ``if` `(j != 0)``                        ``secondway -= prefix[j - 1];` `                    ``int` `answer = count * (count - 1)``                                 ``* secondway * (secondway - 1);``                    ``answer /= 4;``                    ``answer += (pairsum * secondway``                               ``* (secondway - 1)) / 2;` `                    ``// Adding ways for all possible x``                    ``total += answer;``                    ``count = 0;``                ``}``            ``}``        ``}``    ``}` `    ``return` `total;``}` `// Driver code``int` `main()``{``    ``string test = ``"13134422"``;``    ``cout << solve(test) << endl;` `    ``return` `0;``}`

## Java

 `// Java Implementation of above approach``import` `java.io.*;` `class` `GFG``{` `// Function to return the total``// required sub-sequences``static` `int` `solve(String test, ``int` `MOD)``{``    ``int` `size = test.length();``    ``int` `total = ``0``;` `    ``// Find ways for all values of x``    ``for` `(``int` `i = ``0``; i <= ``8``; i++)``    ``{``        ``int` `x = i;` `        ``// x+1``        ``int` `y = i + ``1``;``        ``String newtest = ``""``;` `        ``// Removing all unnecessary digits``        ``for` `(``int` `j = ``0``; j < size; j++)``        ``{``            ``if` `(test.charAt(j) == x + ``48` `|| test.charAt(j) == y + ``48``)``            ``{``                ``newtest += test.charAt(j);``            ``}``        ``}` `        ``if` `(newtest.length() > ``0``) {``            ``int` `size1 = newtest.length();` `            ``// Prefix Sum Array for X+1 digit``            ``int` `[]prefix = ``new` `int``[size1];``            ``for` `(``int` `j = ``0``; j < size1; j++)``            ``{``                ``prefix[j] = ``0``;``                ``if` `(newtest.charAt(j) == y + ``48``)``                ``{``                    ``prefix[j]++;``                ``}``            ``}` `            ``for` `(``int` `j = ``1``; j < size1; j++)``            ``{``                ``prefix[j] += prefix[j - ``1``];``            ``}` `            ``int` `count = ``0``;``            ``int` `firstcount = ``0``;` `            ``// Sum of squares``            ``int` `ss = ``0``;` `            ``// Previous sum of all possible pairs``            ``int` `prev = ``0``;` `            ``for` `(``int` `j = ``0``; j < size1; j++)``            ``{``                ``if` `(newtest.charAt(j) == x + ``48``)``                ``{``                    ``count++;``                    ``firstcount++;``                ``}``                ``else``                ``{` `                    ``ss += count * count;` `                    ``// To find sum of multiplication of all``                    ``// possible pairs``                    ``int` `pairsum = (firstcount * firstcount - ss) / ``2``;``                    ``int` `temp = pairsum;` `                    ``// To prevent overcounting``                    ``pairsum -= prev;``                    ``prev = temp;` `                    ``int` `secondway = prefix[size1 - ``1``];``                    ``if` `(j != ``0``)``                        ``secondway -= prefix[j - ``1``];` `                    ``int` `answer = count * (count - ``1``)``                                ``* secondway * (secondway - ``1``);``                    ``answer /= ``4``;``                    ``answer += (pairsum * secondway``                            ``* (secondway - ``1``)) / ``2``;` `                    ``// Adding ways for all possible x``                    ``total += answer;``                    ``count = ``0``;``                ``}``            ``}``        ``}``    ``}` `    ``return` `total;``}` `// Driver code``public` `static` `void` `main (String[] args)``{``    ``String test = ``"13134422"``;``    ``int` `MOD = ``1000000007``;``    ``System.out.println(solve(test,MOD));` `}``}` `// This code is contributed by krikti..`

## Python3

 `# Python3 implementation of the approach` `MOD``=` `1000000007` `# Function to return the total``# required sub-sequences``def` `solve(test):` `    ``size ``=` `len``(test)``    ``total ``=` `0` `    ``# Find ways for all values of x``    ``for` `i ``in` `range``(``9``):``        ``x ``=` `i` `        ``# x+1``        ``y ``=` `i ``+` `1``        ``newtest``=``""` `        ``# Removing all unnecessary digits``        ``for` `j ``in` `range``(size):``            ``if` `(``ord``(test[j]) ``=``=` `x ``+` `48` `or` `ord``(test[j]) ``=``=` `y ``+` `48``):``                ``newtest ``+``=` `test[j]`  `        ``if` `(``len``(newtest) > ``0``):``            ``size1 ``=` `len``(newtest)` `            ``# Prefix Sum Array for X+1 digit``            ``prefix``=``[``0` `for` `i ``in` `range``(size1)]` `            ``for` `j ``in` `range``(size1):``                ``if` `(``ord``(newtest[j]) ``=``=` `y ``+` `48``):``                    ``prefix[j]``+``=``1` `            ``for` `j ``in` `range``(``1``,size1):``                ``prefix[j] ``+``=` `prefix[j ``-` `1``]` `            ``count ``=` `0``            ``firstcount ``=` `0` `            ``# Sum of squares``            ``ss ``=` `0` `            ``# Previous sum of all possible pairs``            ``prev ``=` `0` `            ``for` `j ``in` `range``(size1):``                ``if` `(``ord``(newtest[j]) ``=``=` `x ``+` `48``):``                    ``count``+``=``1``                    ``firstcount``+``=``1` `                ``else``:` `                    ``ss ``+``=` `count ``*` `count` `                    ``# To find sum of multiplication of all``                    ``# possible pairs``                    ``pairsum ``=` `(firstcount ``*` `firstcount ``-` `ss) ``/``/` `2``                    ``temp ``=` `pairsum` `                    ``# To prevent overcounting``                    ``pairsum ``-``=` `prev``                    ``prev ``=` `temp` `                    ``secondway ``=` `prefix[size1 ``-` `1``]``                    ``if` `(j !``=` `0``):``                        ``secondway ``-``=` `prefix[j ``-` `1``]` `                    ``answer ``=` `count ``*` `(count ``-` `1``)``*` `secondway ``*` `(secondway ``-` `1``)``                    ``answer ``/``/``=` `4``                    ``answer ``+``=` `(pairsum ``*` `secondway ``*` `(secondway ``-` `1``)) ``/``/` `2` `                    ``# Adding ways for all possible x``                    ``total ``+``=` `answer``                    ``count ``=` `0` `    ``return` `total` `# Driver code``test ``=` `"13134422"``print``(solve(test))` `# This code is contributed by mohit kumar 29`

## C#

 `// C# Implementation of above approach` `using` `System;` `class` `GFG``{` `    ``// Function to return the total``    ``// required sub-sequences``    ``static` `int` `solve(``string` `test, ``int` `MOD)``    ``{``        ``int` `size = test.Length;``        ``int` `total = 0;``    ` `        ``// Find ways for all values of x``        ``for` `(``int` `i = 0; i <= 8; i++)``        ``{``            ``int` `x = i;``    ` `            ``// x+1``            ``int` `y = i + 1;``            ``string` `newtest = ``""``;``    ` `            ``// Removing all unnecessary digits``            ``for` `(``int` `j = 0; j < size; j++)``            ``{``                ``if` `(test[j] == x + 48 || test[j] == y + 48)``                ``{``                    ``newtest += test[j];``                ``}``            ``}``    ` `            ``if` `(newtest.Length > 0) {``                ``int` `size1 = newtest.Length;``    ` `                ``// Prefix Sum Array for X+1 digit``                ``int` `[]prefix = ``new` `int``[size1];``                ``for` `(``int` `j = 0; j < size1; j++)``                ``{``                    ``prefix[j] = 0;``                    ``if` `(newtest[j] == y + 48)``                    ``{``                        ``prefix[j]++;``                    ``}``                ``}``    ` `                ``for` `(``int` `j = 1; j < size1; j++)``                ``{``                    ``prefix[j] += prefix[j - 1];``                ``}``    ` `                ``int` `count = 0;``                ``int` `firstcount = 0;``    ` `                ``// Sum of squares``                ``int` `ss = 0;``    ` `                ``// Previous sum of all possible pairs``                ``int` `prev = 0;``    ` `                ``for` `(``int` `j = 0; j < size1; j++)``                ``{``                    ``if` `(newtest[j] == x + 48)``                    ``{``                        ``count++;``                        ``firstcount++;``                    ``}``                    ``else``                    ``{``    ` `                        ``ss += count * count;``    ` `                        ``// To find sum of multiplication of all``                        ``// possible pairs``                        ``int` `pairsum = (firstcount * firstcount - ss) / 2;``                        ``int` `temp = pairsum;``    ` `                        ``// To prevent overcounting``                        ``pairsum -= prev;``                        ``prev = temp;``    ` `                        ``int` `secondway = prefix[size1 - 1];``                        ``if` `(j != 0)``                            ``secondway -= prefix[j - 1];``    ` `                        ``int` `answer = count * (count - 1)``                                    ``* secondway * (secondway - 1);``                        ``answer /= 4;``                        ``answer += (pairsum * secondway``                                ``* (secondway - 1)) / 2;``    ` `                        ``// Adding ways for all possible x``                        ``total += answer;``                        ``count = 0;``                    ``}``                ``}``            ``}``        ``}``    ` `        ``return` `total;``    ``}``    ` `    ``// Driver code``    ``public` `static` `void` `Main ()``    ``{``        ``string` `test = ``"13134422"``;``        ``int` `MOD = 1000000007;``        ``Console.WriteLine(solve(test,MOD));``    ` `    ``}``}` `// This code is contributed by AnkitRai01`

## Javascript

 ``
Output:
`2`

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