# Color N boxes using M colors such that K boxes have different color from the box on its left

Given **N** number of boxes arranged in a row and **M** number of colors. The task is to find the number of ways to paint those **N** boxes using **M** colors such that there are exactly **K** boxes with a color different from the color of the box on its left. Print this answer modulo **998244353**.**Examples:**

Input:N = 3, M = 3, K = 0Output:3

Since the value of K is zero, no box can have a different color from color of the box on its left. Thus, all boxes should be painted with same color and since there are 3 types of colors, so there are total 3 ways.Input:N = 3, M = 2, K = 1Output:4

Let’s number the colors as 1 and 2. Four possible sequences of painting 3 boxes with 1 box having different color from color of box on its left are (1 2 2), (1 1 2), (2 1 1) (2 2 1)

**Prerequisites :** Dynamic Programming

**Approach:** This problem can be solved using dynamic programming where **dp[i][j]** will denote the number of ways to paint **i** boxes using **M** colors such that there are exactly **j** boxes with a color different from the color of the box on its left. For every current box except **1 ^{st}**, either we can paint the same color as painted on its left box and solve for

**dp[i – 1][j]**or we can paint it with remaining

**M – 1**color and solve for

**dp[i – 1][j – 1]**recursively.

Below is the implementation of the above approach:

## C++

`// CPP Program to Paint N boxes using M` `// colors such that K boxes have color` `// different from color of box on its left` `#include <bits/stdc++.h>` `using` `namespace` `std;` `const` `int` `MOD = 998244353;` `vector<vector<` `int` `>> dp;` `// This function returns the required number` `// of ways where idx is the current index and` `// diff is number of boxes having different` `// color from box on its left` `int` `solve(` `int` `idx, ` `int` `diff, ` `int` `N, ` `int` `M, ` `int` `K)` `{` ` ` `// Base Case` ` ` `if` `(idx > N) {` ` ` `if` `(diff == K)` ` ` `return` `1;` ` ` `return` `0;` ` ` `}` ` ` `// If already computed` ` ` `if` `(dp[idx][ diff] != -1)` ` ` `return` `dp[idx][ diff];` ` ` `// Either paint with same color as` ` ` `// previous one` ` ` `int` `ans = solve(idx + 1, diff, N, M, K);` ` ` `// Or paint with remaining (M - 1)` ` ` `// colors` ` ` `ans = ans % MOD + ((M - 1) % MOD * solve(idx + 1, diff + 1, N, M, K) % MOD) % MOD;` ` ` `return` `dp[idx][ diff] = ans;` `}` `// Driver code` `int` `main()` `{` ` ` `int` `N = 3, M = 3, K = 0;` ` ` `dp = vector<vector<` `int` `>>(N+1,vector<` `int` `>(N+1,-1));` ` ` `// Multiply M since first box can be` ` ` `// painted with any of the M colors and` ` ` `// start solving from 2nd box` ` ` `cout << (M * solve(2, 0, N, M, K)) << endl;` ` ` `return` `0;` `}` |

## Java

`// Java Program to Paint N boxes using M` `// colors such that K boxes have color` `// different from color of box on its left` `class` `GFG` `{` ` ` ` ` `static` `int` `M = ` `1001` `;` ` ` `static` `int` `MOD = ` `998244353` `;` ` ` `static` `int` `[][] dp = ` `new` `int` `[M][M];` ` ` `// This function returns the required number` ` ` `// of ways where idx is the current index and` ` ` `// diff is number of boxes having different` ` ` `// color from box on its left` ` ` `static` `int` `solve(` `int` `idx, ` `int` `diff,` ` ` `int` `N, ` `int` `M, ` `int` `K)` ` ` `{` ` ` `// Base Case` ` ` `if` `(idx > N)` ` ` `{` ` ` `if` `(diff == K)` ` ` `return` `1` `;` ` ` `return` `0` `;` ` ` `}` ` ` `// If already computed` ` ` `if` `(dp[idx][ diff] != -` `1` `)` ` ` `return` `dp[idx][ diff];` ` ` `// Either paint with same color as` ` ` `// previous one` ` ` `int` `ans = solve(idx + ` `1` `, diff, N, M, K);` ` ` `// Or paint with remaining (M - 1)` ` ` `// colors` ` ` `ans += (M - ` `1` `) * solve(idx + ` `1` `,` ` ` `diff + ` `1` `, N, M, K);` ` ` `return` `dp[idx][ diff] = ans % MOD;` ` ` `}` ` ` `// Driver code` ` ` `public` `static` `void` `main (String[] args)` ` ` `{` ` ` `int` `N = ` `3` `, M = ` `3` `, K = ` `0` `;` ` ` `for` `(` `int` `i = ` `0` `; i <= M; i++)` ` ` `for` `(` `int` `j = ` `0` `; j <= M; j++)` ` ` `dp[i][j] = -` `1` `;` ` ` ` ` `// Multiply M since first box can be` ` ` `// painted with any of the M colors and` ` ` `// start solving from 2nd box` ` ` `System.out.println((M * solve(` `2` `, ` `0` `, N, M, K)));` ` ` `}` `}` `// This code is contributed by mits` |

## Python3

`# Python3 Program to Paint N boxes using M` `# colors such that K boxes have color` `# different from color of box on its left` `M ` `=` `1001` `;` `MOD ` `=` `998244353` `;` `dp ` `=` `[[` `-` `1` `]` `*` `M ] ` `*` `M` `# This function returns the required number` `# of ways where idx is the current index and` `# diff is number of boxes having different` `# color from box on its left` `def` `solve(idx, diff, N, M, K) :` ` ` ` ` `# Base Case` ` ` `if` `(idx > N) :` ` ` `if` `(diff ` `=` `=` `K) :` ` ` `return` `1` ` ` `return` `0` ` ` `# If already computed` ` ` `if` `(dp[idx][ diff] !` `=` `-` `1` `) :` ` ` `return` `dp[idx];` ` ` `# Either paint with same color as` ` ` `# previous one` ` ` `ans ` `=` `solve(idx ` `+` `1` `, diff, N, M, K);` ` ` `# Or paint with remaining (M - 1)` ` ` `# colors` ` ` `ans ` `+` `=` `(M ` `-` `1` `) ` `*` `solve(idx ` `+` `1` `, diff ` `+` `1` `, N, M, K);` ` ` `dp[idx][ diff] ` `=` `ans ` `%` `MOD;` ` ` ` ` `return` `dp[idx][ diff]` `# Driver code` `if` `__name__ ` `=` `=` `"__main__"` `:` ` ` `N ` `=` `3` ` ` `M ` `=` `3` ` ` `K ` `=` `0` ` ` `# Multiply M since first box can be` ` ` `# painted with any of the M colors and` ` ` `# start solving from 2nd box` ` ` `print` `(M ` `*` `solve(` `2` `, ` `0` `, N, M, K))` `# This code is contributed by Ryuga` |

## C#

`// C# Program to Paint N boxes using M` `// colors such that K boxes have color` `// different from color of box on its left` `using` `System;` `class` `GFG` `{` ` ` `static` `int` `M = 1001;` `static` `int` `MOD = 998244353;` `static` `int` `[,] dp = ` `new` `int` `[M, M];` `// This function returns the required number` `// of ways where idx is the current index and` `// diff is number of boxes having different` `// color from box on its left` `static` `int` `solve(` `int` `idx, ` `int` `diff,` ` ` `int` `N, ` `int` `M, ` `int` `K)` `{` ` ` `// Base Case` ` ` `if` `(idx > N)` ` ` `{` ` ` `if` `(diff == K)` ` ` `return` `1;` ` ` `return` `0;` ` ` `}` ` ` `// If already computed` ` ` `if` `(dp[idx, diff] != -1)` ` ` `return` `dp[idx, diff];` ` ` `// Either paint with same color as` ` ` `// previous one` ` ` `int` `ans = solve(idx + 1, diff, N, M, K);` ` ` `// Or paint with remaining (M - 1)` ` ` `// colors` ` ` `ans += (M - 1) * solve(idx + 1,` ` ` `diff + 1, N, M, K);` ` ` `return` `dp[idx, diff] = ans % MOD;` `}` `// Driver code` `public` `static` `void` `Main ()` `{` ` ` `int` `N = 3, M = 3, K = 0;` ` ` `for` `(` `int` `i = 0; i <= M; i++)` ` ` `for` `(` `int` `j = 0; j <= M; j++)` ` ` `dp[i, j] = -1;` ` ` `// Multiply M since first box can be` ` ` `// painted with any of the M colors and` ` ` `// start solving from 2nd box` ` ` `Console.WriteLine((M * solve(2, 0, N, M, K)));` `}` `}` `// This code is contributed by chandan_jnu` |

## PHP

`<?php` `// PHP Program to Paint N boxes using M` `// colors such that K boxes have color` `// different from color of box on its left` `$M` `= 1001;` `$MOD` `= 998244353;` `$dp` `= ` `array_fill` `(0, ` `$M` `,` ` ` `array_fill` `(0, ` `$M` `, -1));` `// This function returns the required number` `// of ways where idx is the current index` `// and diff is number of boxes having` `// different color from box on its left` `function` `solve(` `$idx` `, ` `$diff` `, ` `$N` `, ` `$M` `, ` `$K` `)` `{` ` ` `global` `$dp` `, ` `$MOD` `;` ` ` ` ` `// Base Case` ` ` `if` `(` `$idx` `> ` `$N` `)` ` ` `{` ` ` `if` `(` `$diff` `== ` `$K` `)` ` ` `return` `1;` ` ` `return` `0;` ` ` `}` ` ` `// If already computed` ` ` `if` `(` `$dp` `[` `$idx` `][` `$diff` `] != -1)` ` ` `return` `$dp` `[` `$idx` `][` `$diff` `];` ` ` `// Either paint with same color` ` ` `// as previous one` ` ` `$ans` `= solve(` `$idx` `+ 1, ` `$diff` `, ` `$N` `, ` `$M` `, ` `$K` `);` ` ` `// Or paint with remaining (M - 1)` ` ` `// colors` ` ` `$ans` `+= (` `$M` `- 1) * solve(` `$idx` `+ 1,` ` ` `$diff` `+ 1, ` `$N` `, ` `$M` `, ` `$K` `);` ` ` `return` `$dp` `[` `$idx` `][` `$diff` `] = ` `$ans` `% ` `$MOD` `;` `}` `// Driver code` `$N` `= 3;` `$M` `= 3;` `$K` `= 0;` `// Multiply M since first box can be` `// painted with any of the M colors and` `// start solving from 2nd box` `echo` `(` `$M` `* solve(2, 0, ` `$N` `, ` `$M` `, ` `$K` `));` `// This code is contributed by chandan_jnu` `?>` |

## Javascript

`<script>` ` ` `// JavaScript Program to Paint N boxes using M` ` ` `// colors such that K boxes have color` ` ` `// different from color of box on its left` ` ` ` ` `let m = 1001;` ` ` `let MOD = 998244353;` ` ` ` ` `let dp = ` `new` `Array(m);` ` ` `for` `(let i = 0; i < m; i++)` ` ` `{` ` ` `dp[i] = ` `new` `Array(m);` ` ` `for` `(let j = 0; j < m; j++)` ` ` `{` ` ` `dp[i][j] = 0;` ` ` `}` ` ` `}` ` ` ` ` `// This function returns the required number` ` ` `// of ways where idx is the current index and` ` ` `// diff is number of boxes having different` ` ` `// color from box on its left` ` ` `function` `solve(idx, diff, N, M, K)` ` ` `{` ` ` `// Base Case` ` ` `if` `(idx > N)` ` ` `{` ` ` `if` `(diff == K)` ` ` `return` `1;` ` ` `return` `0;` ` ` `}` ` ` ` ` `// If already computed` ` ` `if` `(dp[idx][ diff] != -1)` ` ` `return` `dp[idx][ diff];` ` ` ` ` `// Either paint with same color as` ` ` `// previous one` ` ` `let ans = solve(idx + 1, diff, N, M, K);` ` ` ` ` `// Or paint with remaining (M - 1)` ` ` `// colors` ` ` `ans += (M - 1) * solve(idx + 1,` ` ` `diff + 1, N, M, K);` ` ` `dp[idx][ diff] = ans % MOD;` ` ` `return` `dp[idx][ diff];` ` ` `}` ` ` ` ` `let N = 3, M = 3, K = 0;` ` ` `for` `(let i = 0; i <= M; i++)` ` ` `for` `(let j = 0; j <= M; j++)` ` ` `dp[i][j] = -1;` ` ` `// Multiply M since first box can be` ` ` `// painted with any of the M colors and` ` ` `// start solving from 2nd box` ` ` `document.write((M * solve(2, 0, N, M, K)));` ` ` `</script>` |

**Output:**

3

**Time Complexity:** O(M*M)**Auxiliary Space: **O(M*M)