# Number of strings of length N with no palindromic sub string

Given two positive integers N, M. The task is to find the number of strings of length N under the alphabet set of size M such that no substrings of size greater than 1 is palindromic.

Examples:

Input : N = 2, M = 3
Output : 6
In this case, set of alphabet are 3, say {A, B, C}
All possible string of length 2, using 3 letters are:
{AA, AB, AC, BA, BB, BC, CA, CB, CC}
Out of these {AA, BB, CC} contain palindromic substring,
8 - 2 = 6.

Input : N = 2, M = 2
Output : 2
Out of {AA, BB, AB, BA}, only {AB, BA} contain
non-palindromic substrings.

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

First, observe, a string does not contain any palindromic substring if the string doesn’t have any palindromic substring of the length 2 and 3, because all the palindromic string of the greater lengths contains at least one palindromic substring of the length of 2 or 3, basically in the center.

So, the following is true:

• There are M ways to choose the first symbol of the string.
• Then there are (M – 1) ways to choose the second symbol of the string. Basically, it should not be equal to first one.
• Then there are (M – 2) ways to choose any next symbol. Basically, it should not coincide with the previous symbols, that aren’t equal.

Knowing this, we can evaluate the answer in the following ways:

• If N = 1, then the answer will be M.
• If N = 2, then the answer is M*(M – 1).
• If N >= 3, then M * (M – 1) * (M – 2)N-2.

Below is the implementation of above idea :

## C++

 // CPP program to count number of strings of // size m such that no substring is palindrome. #include using namespace std;    // Return the count of strings with // no palindromic substring. int numofstring(int n, int m) {         if (n == 1)         return m;        if (n == 2)         return m * (m - 1);        return m * (m - 1) * pow(m - 2, n - 2); }    // Driven Program int main() {         int n = 2, m = 3;     cout << numofstring(n, m) << endl;     return 0; }

## Java

 // Java program to count number of strings of // size m such that no substring is palindrome. import java.io.*;    class GFG {            // Return the count of strings with     // no palindromic substring.     static int numofstring(int n, int m)     {          if (n == 1)             return m;                if (n == 2)             return m * (m - 1);                return m * (m - 1) * (int)Math.pow(m - 2, n - 2);     }            // Driven Program     public static void main (String[] args)      {         int n = 2, m = 3;         System.out.println(numofstring(n, m));     } }    // This code is contributed by ajit.

## Python3

 # Python3 program to count number of strings of # size m such that no substring is palindrome    # Return the count of strings with # no palindromic substring. def numofstring(n, m):     if n == 1:         return m        if n == 2:         return m * (m - 1)        return m * (m - 1) * pow(m - 2, n - 2)    # Driven Program n = 2 m = 3 print (numofstring(n, m))    # This code is contributed # by Shreyanshi Arun.

## C#

 // C# program to count number of strings of // size m such that no substring is palindrome. using System;    class GFG {            // Return the count of strings with     // no palindromic substring.     static int numofstring(int n, int m)     {          if (n == 1)             return m;                if (n == 2)             return m * (m - 1);                return m * (m - 1) * (int)Math.Pow(m - 2,                                            n - 2);     }            // Driver Code     public static void Main ()      {         int n = 2, m = 3;         Console.Write(numofstring(n, m));     } }    // This code is contributed by Nitin Mittal.

## PHP



Output
6

GeeksforGeeks has prepared a complete interview preparation course with premium videos, theory, practice problems, TA support and many more features. Please refer Placement 100 for details

My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

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.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Improved By : nitin mittal