# 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`

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Improved By : nitin mittal