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

Asked in : Amazon

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 C++ implementation of the idea.

// 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; }

Output:

3

Time Complexity : O(n)

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