# Number of subsequences of the form a^i b^j c^k

Given a string, count number of subsequences of the form a^{i}b^{j}c^{k}, i.e., it consists of i ’a’ characters, followed by j ’b’ characters, followed by k ’c’ characters where i >= 1, j >=1 and k >= 1.

**Note:** Two subsequences are considered different if the set of array indexes picked for the 2 subsequences are different.

Expected Time Complexity : O(n)

**Examples:**

Input : abbc Output : 3 Subsequences are abc, abc and abbc Input : abcabc Output : 7 Subsequences are abc, abc, abbc, aabc abcc, abc and abc

We traverse given string. For every character encounter, we do following:

**1)** Initialize counts of different subsequences caused by different combination of ‘a’. Let this count be aCount.

**2)** Initialize counts of different subsequences caused by different combination of ‘b’. Let this count be bCount.

**3)** Initialize counts of different subsequences caused by different combination of ‘c’. Let this count be cCount.

**4)** Traverse all characters of given string. Do following for current character **s[i]**

**If current character is ‘a’**, then there are following possibilities :

a) Current character begins a new subsequence.

b) Current character is part of aCount subsequences.

c) Current character is not part of aCount subsequences.

Therefore we do aCount = (1 + 2 * aCount);

**If current character is ‘b’**, then there are following possibilities :

a) Current character begins a new subsequence of b’s with aCount subsequences.

b) Current character is part of bCount subsequences.

c) Current character is not part of bCount subsequences.

Therefore we do bCount = (aCount + 2 * bCount);

**If current character is ‘c’**, then there are following possibilities :

a) Current character begins a new subsequence of c’s with bCount subsequences.

b) Current character is part of cCount subsequences.

c) Current character is not part of cCount subsequences.

Therefore we do cCount = (bCount + 2 * cCount);

**5)** Finally we return cCount;

Below is the implementation of above idea :

## C++

`// C++ program to count subsequences of the ` `// form a^i b^j c^k ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Returns count of subsequences of the form ` `// a^i b^j c^k ` `int` `countSubsequences(string s) ` `{ ` ` ` `// Initialize counts of different subsequences ` ` ` `// caused by different combination of 'a' ` ` ` `int` `aCount = 0; ` ` ` ` ` `// Initialize counts of different subsequences ` ` ` `// caused by different combination of 'a' and ` ` ` `// different combination of 'b' ` ` ` `int` `bCount = 0; ` ` ` ` ` `// Initialize counts of different subsequences ` ` ` `// caused by different combination of 'a', 'b' ` ` ` `// and 'c'. ` ` ` `int` `cCount = 0; ` ` ` ` ` `// Traverse all characters of given string ` ` ` `for` `(unsigned ` `int` `i=0; i<s.size(); i++) ` ` ` `{ ` ` ` `/* If current character is 'a', then ` ` ` `there are following possibilities : ` ` ` `a) Current character begins a new ` ` ` `subsequence. ` ` ` `b) Current character is part of aCount ` ` ` `subsequences. ` ` ` `c) Current character is not part of ` ` ` `aCount subsequences. */` ` ` `if` `(s[i] == ` `'a'` `) ` ` ` `aCount = (1 + 2 * aCount); ` ` ` ` ` `/* If current character is 'b', then ` ` ` `there are following possibilities : ` ` ` `a) Current character begins a new ` ` ` `subsequence of b's with aCount ` ` ` `subsequences. ` ` ` `b) Current character is part of bCount ` ` ` `subsequences. ` ` ` `c) Current character is not part of ` ` ` `bCount subsequences. */` ` ` `else` `if` `(s[i] == ` `'b'` `) ` ` ` `bCount = (aCount + 2 * bCount); ` ` ` ` ` `/* If current character is 'c', then ` ` ` `there are following possibilities : ` ` ` `a) Current character begins a new ` ` ` `subsequence of c's with bCount ` ` ` `subsequences. ` ` ` `b) Current character is part of cCount ` ` ` `subsequences. ` ` ` `c) Current character is not part of ` ` ` `cCount subsequences. */` ` ` `else` `if` `(s[i] == ` `'c'` `) ` ` ` `cCount = (bCount + 2 * cCount); ` ` ` `} ` ` ` ` ` `return` `cCount; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `string s = ` `"abbc"` `; ` ` ` `cout << countSubsequences(s) << endl; ` ` ` `return` `0; ` `}` |

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## Java

`// Java program to count subsequences of the ` `// form a^i b^j c^k ` `public` `class` `No_of_subsequence { ` ` ` ` ` `// Returns count of subsequences of the form ` ` ` `// a^i b^j c^k ` ` ` `static` `int` `countSubsequences(String s) ` ` ` `{ ` ` ` `// Initialize counts of different subsequences ` ` ` `// caused by different combination of 'a' ` ` ` `int` `aCount = ` `0` `; ` ` ` ` ` `// Initialize counts of different subsequences ` ` ` `// caused by different combination of 'a' and ` ` ` `// different combination of 'b' ` ` ` `int` `bCount = ` `0` `; ` ` ` ` ` `// Initialize counts of different subsequences ` ` ` `// caused by different combination of 'a', 'b' ` ` ` `// and 'c'. ` ` ` `int` `cCount = ` `0` `; ` ` ` ` ` `// Traverse all characters of given string ` ` ` `for` `(` `int` `i=` `0` `; i< s.length(); i++) ` ` ` `{ ` ` ` `/* If current character is 'a', then ` ` ` `there are following possibilities : ` ` ` `a) Current character begins a new ` ` ` `subsequence. ` ` ` `b) Current character is part of aCount ` ` ` `subsequences. ` ` ` `c) Current character is not part of ` ` ` `aCount subsequences. */` ` ` `if` `(s.charAt(i) == ` `'a'` `) ` ` ` `aCount = (` `1` `+ ` `2` `* aCount); ` ` ` ` ` `/* If current character is 'b', then ` ` ` `there are following possibilities : ` ` ` `a) Current character begins a new ` ` ` `subsequence of b's with aCount ` ` ` `subsequences. ` ` ` `b) Current character is part of bCount ` ` ` `subsequences. ` ` ` `c) Current character is not part of ` ` ` `bCount subsequences. */` ` ` `else` `if` `(s.charAt(i) == ` `'b'` `) ` ` ` `bCount = (aCount + ` `2` `* bCount); ` ` ` ` ` `/* If current character is 'c', then ` ` ` `there are following possibilities : ` ` ` `a) Current character begins a new ` ` ` `subsequence of c's with bCount ` ` ` `subsequences. ` ` ` `b) Current character is part of cCount ` ` ` `subsequences. ` ` ` `c) Current character is not part of ` ` ` `cCount subsequences. */` ` ` `else` `if` `(s.charAt(i) == ` `'c'` `) ` ` ` `cCount = (bCount + ` `2` `* cCount); ` ` ` `} ` ` ` ` ` `return` `cCount; ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `main(String args[]) ` ` ` `{ ` ` ` `String s = ` `"abbc"` `; ` ` ` `System.out.println(countSubsequences(s)); ` ` ` `} ` `} ` `// This code is contributed by Sumit Ghosh ` |

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## Python 3

`# Python 3 program to count ` `# subsequences of the form ` `# a^i b^j c^k ` ` ` `# Returns count of subsequences ` `# of the form a^i b^j c^k ` `def` `countSubsequences(s): ` ` ` ` ` `# Initialize counts of different ` ` ` `# subsequences caused by different ` ` ` `# combination of 'a' ` ` ` `aCount ` `=` `0` ` ` ` ` `# Initialize counts of different ` ` ` `# subsequences caused by different ` ` ` `# combination of 'a' and different ` ` ` `# combination of 'b' ` ` ` `bCount ` `=` `0` ` ` ` ` `# Initialize counts of different ` ` ` `# subsequences caused by different ` ` ` `# combination of 'a', 'b' and 'c'. ` ` ` `cCount ` `=` `0` ` ` ` ` `# Traverse all characters ` ` ` `# of given string ` ` ` `for` `i ` `in` `range` `(` `len` `(s)): ` ` ` ` ` `# If current character is 'a', ` ` ` `# then there are following ` ` ` `# possibilities : ` ` ` `# a) Current character begins ` ` ` `# a new subsequence. ` ` ` `# b) Current character is part ` ` ` `# of aCount subsequences. ` ` ` `# c) Current character is not ` ` ` `# part of aCount subsequences. ` ` ` `if` `(s[i] ` `=` `=` `'a'` `): ` ` ` `aCount ` `=` `(` `1` `+` `2` `*` `aCount) ` ` ` ` ` `# If current character is 'b', then ` ` ` `# there are following possibilities : ` ` ` `# a) Current character begins a ` ` ` `# new subsequence of b's with ` ` ` `# aCount subsequences. ` ` ` `# b) Current character is part ` ` ` `# of bCount subsequences. ` ` ` `# c) Current character is not ` ` ` `# part of bCount subsequences. ` ` ` `elif` `(s[i] ` `=` `=` `'b'` `): ` ` ` `bCount ` `=` `(aCount ` `+` `2` `*` `bCount) ` ` ` ` ` `# If current character is 'c', then ` ` ` `# there are following possibilities : ` ` ` `# a) Current character begins a ` ` ` `# new subsequence of c's with ` ` ` `# bCount subsequences. ` ` ` `# b) Current character is part ` ` ` `# of cCount subsequences. ` ` ` `# c) Current character is not ` ` ` `# part of cCount subsequences. ` ` ` `elif` `(s[i] ` `=` `=` `'c'` `): ` ` ` `cCount ` `=` `(bCount ` `+` `2` `*` `cCount) ` ` ` ` ` `return` `cCount ` ` ` `# Driver code ` `if` `__name__ ` `=` `=` `"__main__"` `: ` ` ` `s ` `=` `"abbc"` ` ` `print` `(countSubsequences(s)) ` ` ` `# This code is contributed ` `# by ChitraNayal ` |

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## C#

`// C# program to count subsequences ` `// of the form a^i b^j c^k ` `using` `System; ` ` ` `public` `class` `GFG { ` ` ` ` ` `// Returns count of subsequences ` ` ` `// of the form a^i b^j c^k ` ` ` `static` `int` `countSubsequences(String s) ` ` ` `{ ` ` ` `// Initialize counts of different ` ` ` `// subsequences caused by different ` ` ` `// combination of 'a' ` ` ` `int` `aCount = 0; ` ` ` ` ` `// Initialize counts of different ` ` ` `// subsequences caused by different ` ` ` `// combination of 'a' and ` ` ` `// different combination of 'b' ` ` ` `int` `bCount = 0; ` ` ` ` ` `// Initialize counts of different ` ` ` `// subsequences caused by different ` ` ` `// combination of 'a', 'b' and 'c' ` ` ` `int` `cCount = 0; ` ` ` ` ` `// Traverse all characters of given string ` ` ` `for` `(` `int` `i = 0; i < s.Length; i++) ` ` ` `{ ` ` ` ` ` `// If current character is 'a', then ` ` ` `// there are following possibilities : ` ` ` `// a) Current character begins a ` ` ` `// new subsequence. ` ` ` `// b) Current character is part ` ` ` `// of aCount subsequences ` ` ` `// c) Current character is not part ` ` ` `// of aCount subsequences. ` ` ` ` ` `if` `(s[i] == ` `'a'` `) ` ` ` `aCount = (1 + 2 * aCount); ` ` ` ` ` `// If current character is 'b', then ` ` ` `// there are following possibilities : ` ` ` `// a) Current character begins a new ` ` ` `// subsequence of b's with aCount ` ` ` `// subsequences. ` ` ` `// b) Current character is part of bCount ` ` ` `// subsequences. ` ` ` `// c) Current character is not part of ` ` ` `// bCount subsequences. ` ` ` `else` `if` `(s[i] == ` `'b'` `) ` ` ` `bCount = (aCount + 2 * bCount); ` ` ` ` ` `// If current character is 'c', then ` ` ` `// there are following possibilities : ` ` ` `// a) Current character begins a new ` ` ` `// subsequence of c's with bCount ` ` ` `// subsequences. ` ` ` `// b) Current character is part of cCount ` ` ` `// subsequences. ` ` ` `// c) Current character is not part of ` ` ` `// cCount subsequences. ` ` ` `else` `if` `(s[i] == ` `'c'` `) ` ` ` `cCount = (bCount + 2 * cCount); ` ` ` `} ` ` ` ` ` `return` `cCount; ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `Main() ` ` ` `{ ` ` ` `String s = ` `"abbc"` `; ` ` ` `Console.Write(countSubsequences(s)); ` ` ` `} ` `} ` ` ` `// This code is contributed by Nitin Mittal. ` |

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## PHP

`<?php ` `// PHP program to count subsequences ` `// of the form a^i b^j c^k ` ` ` `// Returns count of subsequences ` `// of the form a^i b^j c^k ` `function` `countSubsequences(` `$s` `) ` `{ ` ` ` ` ` `// Initialize counts of ` ` ` `// different subsequences ` ` ` `// caused by different ` ` ` `// combination of 'a' ` ` ` `$aCount` `= 0; ` ` ` ` ` `// Initialize counts of ` ` ` `// different subsequences ` ` ` `// caused by different ` ` ` `// combination of 'a' and ` ` ` `// different combination of 'b' ` ` ` `$bCount` `= 0; ` ` ` ` ` `// Initialize counts of ` ` ` `// different subsequences ` ` ` `// caused by different ` ` ` `// combination of 'a', 'b' ` ` ` `// and 'c'. ` ` ` `$cCount` `= 0; ` ` ` ` ` `// Traverse all characters ` ` ` `// of given string ` ` ` `for` `(` `$i` `= 0; ` `$i` `< ` `strlen` `(` `$s` `); ` `$i` `++) ` ` ` `{ ` ` ` ` ` `/* If current character is 'a', then ` ` ` `there are following possibilities : ` ` ` `a) Current character begins a new ` ` ` `subsequence. ` ` ` `b) Current character is part of aCount ` ` ` `subsequences. ` ` ` `c) Current character is not part of ` ` ` `aCount subsequences. */` ` ` `if` `(` `$s` `[` `$i` `] == ` `'a'` `) ` ` ` `$aCount` `= (1 + 2 * ` `$aCount` `); ` ` ` ` ` `/* If current character is 'b', then ` ` ` `there are following possibilities : ` ` ` `a) Current character begins a new ` ` ` `subsequence of b's with aCount ` ` ` `subsequences. ` ` ` `b) Current character is part of bCount ` ` ` `subsequences. ` ` ` `c) Current character is not part of ` ` ` `bCount subsequences. */` ` ` `else` `if` `(` `$s` `[` `$i` `] == ` `'b'` `) ` ` ` `$bCount` `= (` `$aCount` `+ 2 * ` `$bCount` `); ` ` ` ` ` `/* If current character is 'c', then ` ` ` `there are following possibilities : ` ` ` `a) Current character begins a new ` ` ` `subsequence of c's with bCount ` ` ` `subsequences. ` ` ` `b) Current character is part of cCount ` ` ` `subsequences. ` ` ` `c) Current character is not part of ` ` ` `cCount subsequences. */` ` ` `else` `if` `(` `$s` `[` `$i` `] == ` `'c'` `) ` ` ` `$cCount` `= (` `$bCount` `+ 2 * ` `$cCount` `); ` ` ` `} ` ` ` ` ` `return` `$cCount` `; ` `} ` ` ` ` ` `// Driver Code ` ` ` `$s` `= ` `"abbc"` `; ` ` ` `echo` `countSubsequences(` `$s` `) ; ` ` ` `// This code is contributed by nitin mittal. ` `?> ` |

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Output:

3

Time Complexity : O(n)

This article is contributed by **Mr. Somesh Awasthi**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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