# Number of subsequences as “ab” in a string repeated K times

Given a String S, consider a new string formed by repeating the S exactly K times. We need find the number of subsequences as “ab” in the newly formed string.

**Examples :**

Input : S = "abcb" K = 2 Output : 6 Here, Given string is repeated 2 times and we get a new string "abcbabcb" Below are 6 occurrences of "ab"abcbabcbabcbabcbabcbabcbabcbabcbabcbabcb abcbabcbInput : S = "aacbd" K = 1 Output : 2

**Naive Approach**: Finding no.of subsequences of “ab” is in fact finding a pair s[i], s[j] (i < j) where s[i] = ‘a’, s[j] = ‘b’.

We can do this by using two nested for loops and count the no. of pairs.

We can improve this approach in a single traversal of the string. Let us consider an index j, **s[j] =’b’**, if we find no.of index i‘s such that **s[i] = ‘a’** and i < j, then it is same as no.of subsequences of “ab” till j. This can be done by maintaining count of a’s by traversing the array and add the count to our answer at position where s[i] =’b .

**Implementation:-**

## C++

`// CPP code to find number of subsequences of` `// "ab" in the string S which is repeated K times.` `#include <bits/stdc++.h>` `using` `namespace` `std;` `int` `countOccurrences(string s, ` `int` `K)` `{` ` ` `//to store answer` ` ` `int` `ans=0;` ` ` ` ` `//Count of a` ` ` `int` `c=0;` ` ` ` ` `string temp=` `""` `;` ` ` ` ` `//repeating s K times` ` ` `for` `(` `int` `i=0;i<K;i++)` ` ` `{` ` ` `temp+=s;` ` ` `}` ` ` ` ` `//iterating over temp` ` ` `for` `(` `int` `i=0;i<temp.length();i++)` ` ` `{` ` ` `if` `(temp[i]==` `'a'` `)c++;` ` ` ` ` `//if found b then ab can be formed c times for that b` ` ` `//because c times a is present before b` ` ` `else` `if` `(temp[i]==` `'b'` `)ans+=c;` ` ` `}` ` ` `return` `ans;` ` ` `}` `// Driver code` `int` `main()` `{` ` ` `string S = ` `"abcb"` `;` ` ` `int` `k = 2;` ` ` `cout << countOccurrences(S, k) << endl;` ` ` `return` `0;` `}` `//Code contributed by shubhamrajput6156` |

**Output**

6

**Time Complexity**:O(|S|*K)

**Auxiliary Space :- **O(|S|*K)

**Efficient Approach:**

Let T be the newly formed string

T = s1 + s2 + s3 + ….. + sk;

where si is the ith occurrence of the string s.

Here, occurrence of “ab” in T are as follows:

1)”ab” lies completely in the some of occurrence of string S, so we can simply find occurrences of “ab” in Si.Let it be C. So, total no.of occurrences of “ab” in T will be **C*K**.

2) Otherwise, ”a” lies strictly inside some string Si and “b” lies inside some other string Sj, (i < j). In this way finding no.of occurrences of “ab” will be choosing two occurrences of string S from K occurrences**( _{KC2)}** and multiplying it with no. of occurrences of “a” in Si and no.of occurrences of “b” in Sj.

As, Si = Sj = S.

**Time Complexity:** O(|S|), for counting no.of “a”s and no.of “b”s.

## C++

`// CPP code to find number of subsequences of` `// "ab" in the string S which is repeated K times.` `#include <bits/stdc++.h>` `using` `namespace` `std;` `int` `countOccurrences(string s, ` `int` `K)` `{` ` ` `int` `n = s.length();` ` ` `int` `C, c1 = 0, c2 = 0;` ` ` `for` `(` `int` `i = 0; i < n; i++) {` ` ` `if` `(s[i] == ` `'a'` `)` ` ` `c1++; ` `// Count of 'a's` ` ` `if` `(s[i] == ` `'b'` `) {` ` ` `c2++; ` `// Count of 'b's` ` ` `C += c1; ` `// occurrence of "ab"s in string S` ` ` `}` ` ` `}` ` ` `// Add following two :` ` ` `// 1) K * (Occurrences of "ab" in single string)` ` ` `// 2) a is from one string and b is from other.` ` ` `return` `C * K + (K * (K - 1) / 2) * c1 * c2;` `}` `// Driver code` `int` `main()` `{` ` ` `string S = ` `"abcb"` `;` ` ` `int` `k = 2;` ` ` `cout << countOccurrences(S, k) << endl;` ` ` `return` `0;` `}` |

## Java

`// Java code to find number of subsequences of` `// "ab" in the string S which is repeated K times.` `import` `java.io.*;` `class` `GFG {` ` ` `static` `int` `countOccurrences(String s, ` `int` `K)` ` ` `{` ` ` `int` `n = s.length();` ` ` `int` `C = ` `0` `, c1 = ` `0` `, c2 = ` `0` `;` ` ` `for` `(` `int` `i = ` `0` `; i < n; i++) {` ` ` `if` `(s.charAt(i) == ` `'a'` `)` ` ` `c1++; ` `// Count of 'a's` ` ` `if` `(s.charAt(i) == ` `'b'` `) {` ` ` `c2++; ` `// Count of 'b's` ` ` `// occurrence of "ab"s` ` ` `// in string S` ` ` `C += c1;` ` ` `}` ` ` `}` ` ` `// Add following two :` ` ` `// 1) K * (Occurrences of "ab" in single string)` ` ` `// 2) a is from one string and b is from other.` ` ` `return` `C * K + (K * (K - ` `1` `) / ` `2` `) * c1 * c2;` ` ` `}` ` ` `// Driver code` ` ` `public` `static` `void` `main(String[] args)` ` ` `{` ` ` `String S = ` `"abcb"` `;` ` ` `int` `k = ` `2` `;` ` ` `System.out.println(countOccurrences(S, k));` ` ` `}` `}` `// This code is contributed by vt_m.` |

## Python3

`# Python3 code to find number of` `# subsequences of "ab" in the` `# string S which is repeated K times.` `def` `countOccurrences (s, K):` ` ` `n ` `=` `len` `(s)` ` ` `c1 ` `=` `0` ` ` `c2 ` `=` `0` ` ` `C ` `=` `0` ` ` `for` `i ` `in` `range` `(n):` ` ` `if` `s[i] ` `=` `=` `'a'` `:` ` ` `c1` `+` `=` `1` `# Count of 'a's` ` ` `if` `s[i] ` `=` `=` `'b'` `:` ` ` `c2` `+` `=` `1` `# Count of 'b's` ` ` ` ` `# occurrence of "ab"s in string S` ` ` `C ` `+` `=` `c1` ` ` ` ` `# Add following two :` ` ` `# 1) K * (Occurrences of "ab" in single string)` ` ` `# 2) a is from one string and b is from other.` ` ` `return` `C ` `*` `K ` `+` `(K ` `*` `(K ` `-` `1` `) ` `/` `2` `) ` `*` `c1 ` `*` `c2` ` ` ` ` `# Driver code` `S ` `=` `"abcb"` `k ` `=` `2` `print` `(countOccurrences(S, k))` `# This code is contributed by "Abhishek Sharma 44"` |

## C#

`// C# code to find number of subsequences` `// of "ab" in the string S which is` `// repeated K times.` `using` `System;` `class` `GFG {` ` ` `static` `int` `countOccurrences(` `string` `s, ` `int` `K)` ` ` `{` ` ` `int` `n = s.Length;` ` ` `int` `C = 0, c1 = 0, c2 = 0;` ` ` `for` `(` `int` `i = 0; i < n; i++) {` ` ` `if` `(s[i] == ` `'a'` `)` ` ` `// Count of 'a's` ` ` `c1++;` ` ` `if` `(s[i] == ` `'b'` `) {` ` ` `// Count of 'b's` ` ` `c2++;` ` ` `// occurrence of "ab"s` ` ` `// in string S` ` ` `C += c1;` ` ` `}` ` ` `}` ` ` `// Add following two :` ` ` `// 1) K * (Occurrences of "ab" in` ` ` `// single string)` ` ` `// 2) a is from one string and b` ` ` `// is from other.` ` ` `return` `C * K + (K * (K - 1) / 2) * c1 * c2;` ` ` `}` ` ` `// Driver code` ` ` `public` `static` `void` `Main()` ` ` `{` ` ` `string` `S = ` `"abcb"` `;` ` ` `int` `k = 2;` ` ` `Console.WriteLine(` ` ` `countOccurrences(S, k));` ` ` `}` `}` `// This code is contributed by vt_m.` |

## PHP

`<?php` `// PHP code to find number of` `// subsequences of "ab" in the` `// string S which is repeated K times.` `function` `countOccurrences(` `$s` `, ` `$K` `)` `{` ` ` `$n` `= ` `strlen` `(` `$s` `);` ` ` `$C` `= 0; ` `$c1` `= 0; ` `$c2` `= 0;` ` ` `for` `(` `$i` `= 0; ` `$i` `< ` `$n` `; ` `$i` `++)` ` ` `{` ` ` `if` `(` `$s` `[` `$i` `] == ` `'a'` `)` ` ` `// Count of 'a's` ` ` `$c1` `++;` ` ` `if` `(` `$s` `[` `$i` `] == ` `'b'` `)` ` ` `{` ` ` `// Count of 'b's` ` ` `$c2` `++;` ` ` ` ` `// occurrence of "ab"s` ` ` `// in string S` ` ` `$C` `= ` `$C` `+ ` `$c1` `;` ` ` `}` ` ` `}` ` ` `// Add following two :` ` ` `// 1) K * (Occurrences of "ab"` ` ` `// in single string)` ` ` `// 2) a is from one string and` ` ` `// b is from other.` ` ` `return` `$C` `* ` `$K` `+ (` `$K` `* (` `$K` `- 1) / 2) *` ` ` `$c1` `* ` `$c2` `;` `}` `// Driver code` `$S` `= ` `"abcb"` `;` `$k` `= 2;` `echo` `countOccurrences(` `$S` `, ` `$k` `), ` `"\n"` `;` `// This code is contributed by ajit.` `?>` |

## Javascript

`<script>` `// JavaScript code to find number of subsequences of` `// "ab" in the string S which is repeated K times.` `function` `countOccurrences(s, K)` `{` ` ` `let n = s.length;` ` ` `let C = 0, c1 = 0, c2 = 0;` ` ` ` ` `for` `(let i = 0; i < n; i++)` ` ` `{` ` ` `if` `(s[i] == ` `'a'` `)` ` ` `c1++; ` `// Count of 'a's` ` ` `if` `(s[i] == ` `'b'` `)` ` ` `{` ` ` `c2++; ` `// Count of 'b's` ` ` `// Occurrence of "ab"s` ` ` `// in string S` ` ` `C += c1;` ` ` `}` ` ` `}` ` ` `// Add following two :` ` ` `// 1) K * (Occurrences of "ab" in single string)` ` ` `// 2) a is from one string and b is from other.` ` ` `return` `C * K + (K * (K - 1) / 2) * c1 * c2;` `}` `// Driver Code` `let S = ` `"abcb"` `;` `let k = 2;` `document.write(countOccurrences(S, k));` `// This code is contributed by susmitakundugoaldanga` `</script>` |

**Output**

6

**Time complexity:**O(|S|).**Auxiliary Space:** O(|S|)), where n is the length of the string. This is because when string is passed to any function it is passed by value and creates a copy of itself in stack.

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