Count of strings that can be formed using a, b and c under given constraints

Difficulty Level : Medium
Last Updated : 13 Apr, 2020

Given a length n, count the number of strings of length n that can be made using ‘a’, ‘b’ and ‘c’ with at-most one ‘b’ and two ‘c’s allowed.

Examples :

Input : n = 3 
Output : 19 
Below strings follow given constraints:
aaa aab aac aba abc aca acb acc baa
bac bca bcc caa cab cac cba cbc cca ccb 

Input  : n = 4
Output : 39

Asked in Google Interview

Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

A simple solution is to recursively count all possible combination of string that can be mode up to latter ‘a’, ‘b’, and ‘c’.

Below is implementation of above idea

C++

// C++ program to count number of strings 
// of n characters with 
#include<bits/stdc++.h> 
using namespace std; 
  
// n is total number of characters. 
// bCount and cCount are counts of 'b' 
// and 'c' respectively. 
int countStr(int n, int bCount, int cCount) 
{ 
    // Base cases 
    if (bCount < 0 || cCount < 0) return 0; 
    if (n == 0) return 1; 
    if (bCount == 0 && cCount == 0) return 1; 
  
    // Three cases, we choose, a or b or c 
    // In all three cases n decreases by 1. 
    int res = countStr(n-1, bCount, cCount); 
    res += countStr(n-1, bCount-1, cCount); 
    res += countStr(n-1, bCount, cCount-1); 
  
    return res; 
} 
  
// Driver code 
int main() 
{ 
    int n = 3;  // Total number of characters 
    cout << countStr(n, 1, 2); 
    return 0; 
} 

Java

// Java program to count number  
// of strings of n characters with 
import java.io.*; 
  
class GFG  
{ 
      
// n is total number of characters. 
// bCount and cCount are counts of 'b' 
// and 'c' respectively. 
static int countStr(int n,  
                    int bCount,  
                    int cCount) 
{ 
    // Base cases 
    if (bCount < 0 || cCount < 0) return 0; 
    if (n == 0) return 1; 
    if (bCount == 0 && cCount == 0) return 1; 
  
    // Three cases, we choose, a or b or c 
    // In all three cases n decreases by 1. 
    int res = countStr(n - 1, bCount, cCount); 
    res += countStr(n - 1, bCount - 1, cCount); 
    res += countStr(n - 1, bCount, cCount - 1); 
  
    return res; 
} 
  
// Driver code 
public static void main (String[] args) 
{ 
    int n = 3; // Total number of characters 
    System.out.println(countStr(n, 1, 2)); 
} 
} 
  
// This code is contributed by akt_mit 

Python 3

# Python 3 program to  
# count number of strings 
# of n characters with 
  
# n is total number of characters. 
# bCount and cCount are counts  
# of 'b' and 'c' respectively. 
def countStr(n, bCount, cCount): 
  
    # Base cases 
    if (bCount < 0 or cCount < 0): 
        return 0
    if (n == 0) : 
        return 1
    if (bCount == 0 and cCount == 0): 
        return 1
  
    # Three cases, we choose, a or b or c 
    # In all three cases n decreases by 1. 
    res = countStr(n - 1, bCount, cCount) 
    res += countStr(n - 1, bCount - 1, cCount) 
    res += countStr(n - 1, bCount, cCount - 1) 
  
    return res 
  
# Driver code 
if __name__ =="__main__": 
    n = 3 # Total number of characters 
    print(countStr(n, 1, 2)) 
  
# This code is contributed  
# by ChitraNayal 

C#

// C# program to count number  
// of strings of n characters  
// with a, b and c under given 
// constraints 
using System; 
  
class GFG 
{ 
      
// n is total number of  
// characters. bCount and 
// cCount are counts of  
// 'b' and 'c' respectively. 
static int countStr(int n,  
                    int bCount,  
                    int cCount) 
{ 
    // Base cases 
    if (bCount < 0 || cCount < 0)  
        return 0; 
    if (n == 0) return 1; 
    if (bCount == 0 && cCount == 0)  
        return 1; 
  
    // Three cases, we choose,  
    // a or b or c. In all three 
    // cases n decreases by 1. 
    int res = countStr(n - 1,  
                       bCount, cCount); 
    res += countStr(n - 1,  
                    bCount - 1, cCount); 
    res += countStr(n - 1,  
                    bCount, cCount - 1); 
  
    return res; 
} 
  
// Driver code 
static public void Main () 
{ 
    // Total number  
    // of characters 
    int n = 3;  
    Console.WriteLine(countStr(n, 1, 2)); 
} 
} 
  
// This code is contributed by aj_36 

PHP

<?php 
// PHP program to count number of  
// strings of n characters with 
  
// n is total number of characters. 
// bCount and cCount are counts  
// of 'b' and 'c' respectively. 
function countStr($n, $bCount,  
                      $cCount) 
{ 
    // Base cases 
    if ($bCount < 0 ||  
        $cCount < 0) 
        return 0; 
    if ($n == 0) 
    return 1; 
    if ($bCount == 0 &&  
        $cCount == 0)  
        return 1; 
  
    // Three cases, we choose, 
    // a or b or c. In all three  
    // cases n decreases by 1. 
    $res = countStr($n - 1,  
                    $bCount,  
                    $cCount); 
    $res += countStr($n - 1,  
                     $bCount - 1,  
                     $cCount); 
    $res += countStr($n - 1,  
                     $bCount,  
                     $cCount - 1); 
  
    return $res; 
} 
  
// Driver code 
$n = 3; // Total number  
        // of characters 
echo countStr($n, 1, 2); 
      
// This code is contributed by ajit 
?> 

Output :

Time complexity of above solution is exponential.

Efficient Solution
If we drown a recursion tree of above code, we can notice that same values appear multiple times. So we store results which are used later if repeated.

C++

// C++ program to count number of strings 
// of n characters with 
#include<bits/stdc++.h> 
using namespace std; 
  
// n is total number of characters. 
// bCount and cCount are counts of 'b' 
// and 'c' respectively. 
int countStrUtil(int dp[][2][3], int n, int bCount=1, 
                 int cCount=2) 
{ 
    // Base cases 
    if (bCount < 0 || cCount < 0) return 0; 
    if (n == 0) return 1; 
    if (bCount == 0 && cCount == 0) return 1; 
  
    // if we had saw this combination previously 
    if (dp[n][bCount][cCount] != -1) 
        return dp[n][bCount][cCount]; 
  
    // Three cases, we choose, a or b or c 
    // In all three cases n decreases by 1. 
    int res = countStrUtil(dp, n-1, bCount, cCount); 
    res += countStrUtil(dp, n-1, bCount-1, cCount); 
    res += countStrUtil(dp, n-1, bCount, cCount-1); 
  
    return (dp[n][bCount][cCount] = res); 
} 
  
// A wrapper over countStrUtil() 
int countStr(int n) 
{ 
    int dp[n+1][2][3]; 
    memset(dp, -1, sizeof(dp)); 
    return countStrUtil(dp, n); 
} 
  
// Driver code 
int main() 
{ 
    int n = 3; // Total number of characters 
    cout << countStr(n); 
    return 0; 
} 

Java

// Java program to count number of strings  
// of n characters with  
  
class GFG  
{ 
    // n is total number of characters.  
    // bCount and cCount are counts of 'b'  
    // and 'c' respectively.  
  
    static int countStrUtil(int[][][] dp, int n, 
                            int bCount, int cCount)  
    { 
  
        // Base cases  
        if (bCount < 0 || cCount < 0)  
        { 
            return 0; 
        } 
        if (n == 0) 
        { 
            return 1; 
        } 
        if (bCount == 0 && cCount == 0)  
        { 
            return 1; 
        } 
  
        // if we had saw this combination previously  
        if (dp[n][bCount][cCount] != -1)  
        { 
            return dp[n][bCount][cCount]; 
        } 
  
        // Three cases, we choose, a or b or c  
        // In all three cases n decreases by 1.  
        int res = countStrUtil(dp, n - 1, bCount, cCount); 
        res += countStrUtil(dp, n - 1, bCount - 1, cCount); 
        res += countStrUtil(dp, n - 1, bCount, cCount - 1); 
  
        return (dp[n][bCount][cCount] = res); 
    } 
  
    // A wrapper over countStrUtil()  
    static int countStr(int n, int bCount, int cCount)  
    { 
        int[][][] dp = new int[n + 1][2][3]; 
        for (int i = 0; i < n + 1; i++)  
        { 
            for (int j = 0; j < 2; j++)  
            { 
                for (int k = 0; k < 3; k++)  
                { 
                    dp[i][j][k] = -1; 
                } 
            } 
        } 
        return countStrUtil(dp, n,bCount,cCount); 
    } 
  
    // Driver code  
    public static void main(String[] args)  
    { 
        int n = 3; // Total number of characters  
        int bCount = 1, cCount = 2; 
        System.out.println(countStr(n,bCount,cCount)); 
    } 
} 
  
// This code has been contributed by 29AjayKumar 

Python3

# Python 3 program to count number of strings 
# of n characters with 
  
# n is total number of characters. 
# bCount and cCount are counts of 'b' 
# and 'c' respectively. 
def countStrUtil(dp, n, bCount=1,cCount=2): 
  
    # Base cases 
    if (bCount < 0 or cCount < 0): 
        return 0
    if (n == 0): 
        return 1
    if (bCount == 0 and cCount == 0): 
        return 1
  
    # if we had saw this combination previously 
    if (dp[n][bCount][cCount] != -1): 
        return dp[n][bCount][cCount] 
  
    # Three cases, we choose, a or b or c 
    # In all three cases n decreases by 1. 
    res = countStrUtil(dp, n-1, bCount, cCount) 
    res += countStrUtil(dp, n-1, bCount-1, cCount) 
    res += countStrUtil(dp, n-1, bCount, cCount-1) 
  
    dp[n][bCount][cCount] = res 
    return dp[n][bCount][cCount] 
  
# A wrapper over countStrUtil() 
def countStr(n): 
  
    dp = [ [ [-1 for x in range(n+2)] for y in range(3)]for z in range(4)] 
    return countStrUtil(dp, n) 
  
# Driver code 
if __name__ == "__main__": 
      
    n = 3 # Total number of characters 
    print(countStr(n)) 
      
# This code is contributed by chitranayal     

C#

// C# program to count number of strings  
// of n characters with  
using System;  
  
class GFG  
{  
    // n is total number of characters.  
    // bCount and cCount are counts of 'b'  
    // and 'c' respectively.  
    static int countStrUtil(int[,,] dp, int n,  
                    int bCount=1, int cCount=2)  
    {  
        // Base cases  
        if (bCount < 0 || cCount < 0) 
            return 0;  
        if (n == 0)  
            return 1;  
        if (bCount == 0 && cCount == 0)  
            return 1;  
      
        // if we had saw this combination previously  
        if (dp[n,bCount,cCount] != -1)  
            return dp[n,bCount,cCount];  
      
        // Three cases, we choose, a or b or c  
        // In all three cases n decreases by 1.  
        int res = countStrUtil(dp, n - 1, bCount, cCount);  
        res += countStrUtil(dp, n - 1, bCount - 1, cCount);  
        res += countStrUtil(dp, n - 1, bCount, cCount - 1);  
      
        return (dp[n, bCount, cCount] = res);  
    }  
      
    // A wrapper over countStrUtil()  
    static int countStr(int n)  
    {  
        int[,,] dp = new int[n + 1, 2, 3];  
        for(int i = 0; i < n + 1; i++) 
            for(int j = 0; j < 2; j++) 
                for(int k = 0; k < 3; k++) 
                    dp[i, j, k] = -1; 
        return countStrUtil(dp, n);  
    }  
      
    // Driver code  
    static void Main()  
    {  
        int n = 3; // Total number of characters  
          
        Console.Write(countStr(n));  
    } 
} 
  
// This code is contributed by DrRoot_ 

Output :

Time Complexity : O(n)
Auxiliary Space : O(n)

Thanks to Mr. Lazy for suggesting above solutions.

A solution that works in O(1) time :

C++

// A O(1) CPP program to find number of strings 
// that can be made under given constraints. 
#include<bits/stdc++.h> 
using namespace std; 
int countStr(int n) 
{ 
    return 1+(n*2)+(n*((n*n)-1)/2); 
} 
  
// Driver code  
int main() 
{ 
  int n = 3; 
  cout << countStr(n); 
  return 0; 
}  

Java

// A O(1) Java program to  
// find number of strings 
// that can be made under 
// given constraints. 
import java.io.*; 
  
class GFG 
{ 
    static int countStr(int n) 
    { 
    return 1 + (n * 2) +  
           (n * ((n * n) - 1) / 2); 
    } 
  
// Driver code  
public static void main (String[] args) 
{ 
    int n = 3; 
    System.out.println( countStr(n)); 
} 
} 
  
// This code is contributed by ajit 

Python 3

# A O(1) Python3 program to find  
# number of strings that can be 
# made under given constraints. 
  
def countStr(n): 
    return (1 + (n * 2) + 
                (n * ((n * n) - 1) // 2)) 
  
# Driver code  
if __name__ == "__main__": 
    n = 3
    print(countStr(n)) 
  
# This code is contributed 
# by ChitraNayal 

C#

// A O(1) C# program to  
// find number of strings 
// that can be made under 
// given constraints. 
using System; 
  
class GFG 
{ 
    static int countStr(int n) 
    { 
    return 1 + (n * 2) +  
          (n * ((n * n) - 1) / 2); 
    } 
  
// Driver code  
static public void Main () 
{ 
    int n = 3; 
    Console.WriteLine(countStr(n)); 
} 
} 
  
// This code is contributed by m_kit 

PHP

<?php 
// A O(1) PHP program to find  
// number of strings that can  
// be made under given constraints. 
function countStr($n) 
{ 
    return 1 + ($n * 2) + ($n *  
              (($n * $n) - 1) / 2); 
} 
  
// Driver code  
$n = 3; 
echo countStr($n); 
  
// This code is contributed by aj_36 
?> 

Output :

Time Complexity : O(1)
Auxiliary Space : O(1)

Thanks to Niharika Sahai for providing above solution.

Reference :
https://careercup.appspot.com/question?id=5717453712654336

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