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Ways to fill N positions using M colors such that there are exactly K pairs of adjacent different colors
• Difficulty Level : Hard
• Last Updated : 21 Jan, 2019

Given three integers N, M and K. The task is to find the number of ways to fill N positions using M colors such that there are exactly K pairs of adjacent different colors.

Examples:

Input: N = 3, M = 2, K = 1
Output: 4
Let the colors be 1 and 2, so the ways are:
1, 1, 2
1, 2, 2
2, 2, 1
2, 1, 1
Above 4 ways have exactly one pair of adjacent elements with different color.

Input: N = 3, M = 3, K = 2
Output: 12

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

Approach: We can use Dynamic Programming with memoisation to solve the above problem. There are N positions to fill, hence the recursive function will compose of two calls, one if the next position is filled with the same color and the other if it is filled with a different color. Hence the recursive calls will be:

• countWays(index + 1, cnt), if the next index is filled with same color.
• (m – 1) * countWays(index + 1, cnt + 1), if the next index is filled with a different color. The number of ways get multiplied with (m – 1).

The bases cases will be:

• If index = n, then a check for the value of cnt is done. If cnt = K then it is a possible way hence return 1, else return 0.
• To avoid repetitive calls, memoize the returned value in a 2-D array and return this value if the recursive call with same parameters is done again.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``using` `namespace` `std;``#define max 4`` ` `// Recursive function to find the required number of ways``int` `countWays(``int` `index, ``int` `cnt, ``int` `dp[][max], ``int` `n, ``int` `m, ``int` `k)``{`` ` `    ``// When all positions are filled``    ``if` `(index == n) {`` ` `        ``// If adjacent pairs are exactly K``        ``if` `(cnt == k)``            ``return` `1;``        ``else``            ``return` `0;``    ``}`` ` `    ``// If already calculated``    ``if` `(dp[index][cnt] != -1)``        ``return` `dp[index][cnt];`` ` `    ``int` `ans = 0;`` ` `    ``// Next position filled with same color``    ``ans += countWays(index + 1, cnt, dp, n, m, k);`` ` `    ``// Next position filled with different color``    ``// So there can be m-1 different colors``    ``ans += (m - 1) * countWays(index + 1, cnt + 1, dp, n, m, k);`` ` `    ``return` `dp[index][cnt] = ans;``}`` ` `// Driver Code``int` `main()``{``    ``int` `n = 3, m = 3, k = 2;``    ``int` `dp[n + 1][max];``    ``memset``(dp, -1, ``sizeof` `dp);`` ` `    ``cout << m * countWays(1, 0, dp, n, m, k);``}`

## Java

 `//Java implementation of the approach``class` `solution``{``static` `final` `int`  `max=``4``;``  ` `// Recursive function to find the required number of ways``static` `int` `countWays(``int` `index, ``int` `cnt, ``int` `dp[][], ``int` `n, ``int` `m, ``int` `k)``{``  ` `    ``// When all positions are filled``    ``if` `(index == n) {``  ` `        ``// If adjacent pairs are exactly K``        ``if` `(cnt == k)``            ``return` `1``;``        ``else``            ``return` `0``;``    ``}``  ` `    ``// If already calculated``    ``if` `(dp[index][cnt] != -``1``)``        ``return` `dp[index][cnt];``  ` `    ``int` `ans = ``0``;``  ` `    ``// Next position filled with same color``    ``ans += countWays(index + ``1``, cnt, dp, n, m, k);``  ` `    ``// Next position filled with different color``    ``// So there can be m-1 different colors``    ``ans += (m - ``1``) * countWays(index + ``1``, cnt + ``1``, dp, n, m, k);``  ` `    ``return` `dp[index][cnt] = ans;``}``  ` `// Driver Code``public` `static` `void` `main(String args[])``{``    ``int` `n = ``3``, m = ``3``, k = ``2``;``    ``int` `dp[][]= ``new` `int` `[n + ``1``][max];``    ``for``(``int` `i=``0``;i

## Python 3

 `# Python 3 implementation of the approach`` ` `max` `=` `4`` ` `# Recursive function to find the ``# required number of ways``def` `countWays(index, cnt, dp, n, m, k):`` ` `    ``# When all positions are filled``    ``if` `(index ``=``=` `n) :`` ` `        ``# If adjacent pairs are exactly K``        ``if` `(cnt ``=``=` `k):``            ``return` `1``        ``else``:``            ``return` `0`` ` `    ``# If already calculated``    ``if` `(dp[index][cnt] !``=` `-``1``):``        ``return` `dp[index][cnt]`` ` `    ``ans ``=` `0`` ` `    ``# Next position filled with same color``    ``ans ``+``=` `countWays(index ``+` `1``, cnt, dp, n, m, k)`` ` `    ``# Next position filled with different color``    ``# So there can be m-1 different colors``    ``ans ``+``=` `(m ``-` `1``) ``*` `countWays(index ``+` `1``, ``                               ``cnt ``+` `1``, dp, n, m, k)`` ` `    ``dp[index][cnt] ``=` `ans``    ``return` `dp[index][cnt]`` ` `# Driver Code``if` `__name__ ``=``=` `"__main__"``:``     ` `    ``n ``=` `3``    ``m ``=` `3``    ``k ``=` `2``    ``dp ``=` `[[``-``1` `for` `x ``in` `range``(n ``+` `1``)] ``              ``for` `y ``in` `range``(``max``)]`` ` `    ``print``(m ``*` `countWays(``1``, ``0``, dp, n, m, k))`` ` `# This code is contributed by ita_c`

## C#

 `// C# implementation of the approach `` ` `using` `System;`` ` `class` `solution ``{ ``static` `int` `max=4; `` ` `// Recursive function to find the required number of ways ``static` `int` `countWays(``int` `index, ``int` `cnt, ``int` `[,]dp, ``int` `n, ``int` `m, ``int` `k) ``{ `` ` `    ``// When all positions are filled ``    ``if` `(index == n) { `` ` `        ``// If adjacent pairs are exactly K ``        ``if` `(cnt == k) ``            ``return` `1; ``        ``else``            ``return` `0; ``    ``} `` ` `    ``// If already calculated ``    ``if` `(dp[index,cnt] != -1) ``        ``return` `dp[index,cnt]; `` ` `    ``int` `ans = 0; `` ` `    ``// Next position filled with same color ``    ``ans += countWays(index + 1, cnt, dp, n, m, k); `` ` `    ``// Next position filled with different color ``    ``// So there can be m-1 different colors ``    ``ans += (m - 1) * countWays(index + 1, cnt + 1, dp, n, m, k); `` ` `    ``return` `dp[index,cnt] = ans; ``} `` ` `// Driver Code ``public` `static` `void` `Main() ``{ ``    ``int` `n = 3, m = 3, k = 2; ``    ``int` `[,]dp= ``new` `int` `[n + 1,max]; ``    ``for``(``int` `i=0;i

## PHP

 ``
Output:
```12
```

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