# Count of arrays having consecutive element with different values

Given three positive integers n, k and x. The task is to count the number of different array that can be formed of size n such that each element is between 1 to k and two consecutive element are different. Also, the first and last elements of each array should be 1 and x respectively.

Examples :

```Input : n = 4, k = 3, x = 2
Output : 3 ```

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

The idea is to use Dynamic Programming and combinatorics to solve the problem.
First of all, notice that the answer is same for all x from 2 to k. It can easily be proved. This will be useful later on.
Let the state f(i) denote the number of ways to fill the range [1, i] of array A such that A1 = 1 and Ai ≠ 1.
Therefore, if x ≠ 1, the answer to the problem is f(n)/(k – 1), because f(n) is the number of way where An is filled with a number from 2 to k, and the answer are equal for all such values An, so the answer for an individual value is f(n)/(k – 1).
Otherwise, if x = 1, the answer is f(n – 1), because An – 1 ≠ 1, and the only number we can fill An with is x = 1.
Now, the main problem is how to calculate f(i). Consider all numbers that Ai – 1 can be. We know that it must lie in [1, k].

• If Ai – 1 ≠ 1, then there are (k – 2)f(i – 1) ways to fill in the rest of the array, because Ai cannot be 1 or Ai – 1 (so we multiply with (k – 2)), and for the range [1, i – 1], there are, recursively, f(i – 1) ways.
• If Ai – 1 = 1, then there are (k – 1)f(i – 2) ways to fill in the rest of the array, because Ai – 1 = 1 means Ai – 2 ≠ 1 which means there are f(i – 2)ways to fill in the range [1, i – 2] and the only value that Ai cannot be 1, so we have (k – 1) choices for Ai.

By combining the above, we get

`f(i) = (k - 1) * f(i - 2) + (k - 2) * f(i - 1)`

This will help us to use dynamic programming using f(i).

Below is the implementation of this approach:

## C++

 `// CPP Program to find count of arrays. ` `#include ` `#define MAXN 109 ` `using` `namespace` `std; ` ` `  `// Return the number of arrays with given constartints. ` `int` `countarray(``int` `n, ``int` `k, ``int` `x) ` `{ ` `    ``int` `dp[MAXN] = { 0 }; ` ` `  `    ``// Initalising dp and dp. ` `    ``dp = 0; ` `    ``dp = 1; ` ` `  `    ``// Computing f(i) for each 2 <= i <= n. ` `    ``for` `(``int` `i = 2; i < n; i++) ` `        ``dp[i] = (k - 2) * dp[i - 1] + ` `                ``(k - 1) * dp[i - 2]; ` ` `  `    ``return` `(x == 1 ? (k - 1) * dp[n - 2] : dp[n - 1]); ` `} ` ` `  `// Driven Program ` `int` `main() ` `{ ` `    ``int` `n = 4, k = 3, x = 2; ` `    ``cout << countarray(n, k, x) << endl; ` `    ``return` `0; ` `} `

## Java

 `// Java program to find count of arrays. ` `import` `java.util.*; ` ` `  `class` `Counting ` `{ ` `    ``static` `int` `MAXN = ``109``; ` ` `  `    ``public` `static` `int` `countarray(``int` `n, ``int` `k,  ` `                                       ``int` `x) ` `    ``{ ` `        ``int``[] dp = ``new` `int``[``109``]; ` ` `  `        ``// Initalising dp and dp. ` `        ``dp[``0``] = ``0``; ` `        ``dp[``1``] = ``1``; ` ` `  `        ``// Computing f(i) for each 2 <= i <= n. ` `        ``for` `(``int` `i = ``2``; i < n; i++) ` `            ``dp[i] = (k - ``2``) * dp[i - ``1``] + ` `                ``(k - ``1``) * dp[i - ``2``]; ` ` `  `        ``return` `(x == ``1` `? (k - ``1``) * dp[n - ``2``] :  ` `                                  ``dp[n - ``1``]); ` `    ``} ` `     `  `    ``// driver code ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `n = ``4``, k = ``3``, x = ``2``; ` `        ``System.out.println(countarray(n, k, x)); ` `    ``} ` `} ` ` `  ` `  `// This code is contributed by rishabh_jain `

## Python3

 `# Python3 code to find count of arrays. ` ` `  `# Return the number of lists with  ` `# given constraints. ` `def` `countarray( n , k , x ): ` `     `  `    ``dp ``=` `list``() ` `     `  `    ``# Initalising dp and dp ` `    ``dp.append(``0``) ` `    ``dp.append(``1``) ` `     `  `    ``# Computing f(i) for each 2 <= i <= n. ` `    ``i ``=` `2` `    ``while` `i < n: ` `        ``dp.append( (k ``-` `2``) ``*` `dp[i ``-` `1``] ``+` `                   ``(k ``-` `1``) ``*` `dp[i ``-` `2``]) ` `        ``i ``=` `i ``+` `1` `     `  `    ``return` `( (k ``-` `1``) ``*` `dp[n ``-` `2``] ``if` `x ``=``=` `1` `else` `dp[n ``-` `1``]) ` ` `  `# Driven code ` `n ``=` `4` `k ``=` `3` `x ``=` `2` `print``(countarray(n, k, x)) ` ` `  `# This code is contributed by "Sharad_Bhardwaj". `

## C#

 `// C# program to find count of arrays. ` `using` `System; ` ` `  `class` `GFG ` `{ ` `// static int MAXN = 109; ` ` `  `    ``public` `static` `int` `countarray(``int` `n, ``int` `k,  ` `                                    ``int` `x) ` `    ``{ ` `        ``int``[] dp = ``new` `int``; ` ` `  `        ``// Initalising dp and dp. ` `        ``dp = 0; ` `        ``dp = 1; ` ` `  `        ``// Computing f(i) for each 2 <= i <= n. ` `        ``for` `(``int` `i = 2; i < n; i++) ` `            ``dp[i] = (k - 2) * dp[i - 1] + ` `                    ``(k - 1) * dp[i - 2]; ` ` `  `        ``return` `(x == 1 ? (k - 1) * dp[n - 2] :  ` `                                   ``dp[n - 1]); ` `    ``} ` `     `  `    ``// Driver code ` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``int` `n = 4, k = 3, x = 2; ` `        ``Console.WriteLine(countarray(n, k, x)); ` `    ``} ` `} ` ` `  ` `  `// This code is contributed by vt_m `

## PHP

 ` `

Output :

```3
```

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