# Number of ways to choose elements from the array such that their average is K

Given an array arr[] of N integers and an integer K. The task is to find the number of ways to select one or more elements from the array such that the average of the selected integers is equal to given number K.

Examples:

Input: arr[] = {7, 9, 8, 9}, K = 8
Output: 5
{8}, {7, 9}, {7, 9}, {7, 8, 9} and {7, 8, 9}

Input: arr[] = {3, 6, 2, 8, 7, 6, 5, 9}, K = 5
Output: 19

Input: arr[] = {6, 6, 9}, K = 8
Output: 0

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

Simple Approach: A simple solution would be to try for all possibilities since N can be large. Time complexity can be 2N.

Efficient Approach: The above approach can be optimized by using dynamic programming to solve this problem. Suppose we are at i_th index, and let val be the current value of that index. We have two possibilities either to choose that element in the answer or discard the element. Hence we are done now. We will also keep track of the number of elements in our current set of chosen elements.

Following is the recursive formula.

```ways(index, sum, count)
= ways(index - 1, sum, count)
+ ways(index - 1, sum + arr[index], count + 1)
```

Below is the implementation of the above approach:

## C++

 `#include ` `using` `namespace` `std; ` ` `  `#define MAX_INDEX 51 ` `#define MAX_SUM 2505 ` ` `  `// This dp array is used to store our values ` `// so that we don't have to calculate same ` `// values again and again ` `int` `dp[MAX_INDEX][MAX_SUM][MAX_INDEX]; ` ` `  `int` `waysutil(``int` `index, ``int` `sum, ``int` `count, ` `             ``vector<``int``>& arr, ``int` `K) ` `{ ` `    ``// Base cases ` `    ``// Index can't be less than 0 ` `    ``if` `(index < 0) ` `        ``return` `0; ` ` `  `    ``if` `(index == 0) { ` ` `  `        ``// No element is picked hence ` `        ``// average cannot be calculated ` `        ``if` `(count == 0) ` `            ``return` `0; ` `        ``int` `remainder = sum % count; ` ` `  `        ``// If remainder is non zero, we cannot ` `        ``// divide the sum by count i.e. the average ` `        ``// will not be an integer ` `        ``if` `(remainder != 0) ` `            ``return` `0; ` `        ``int` `average = sum / count; ` ` `  `        ``// If we find an average return 1 ` `        ``if` `(average == K) ` `            ``return` `1; ` `    ``} ` ` `  `    ``// If we have already calculated this function ` `    ``// simply return it instead of calculating it again ` `    ``if` `(dp[index][sum][count] != -1) ` `        ``return` `dp[index][sum][count]; ` ` `  `    ``// If we don't pick the current element ` `    ``// simple recur for index -1 ` `    ``int` `dontpick = waysutil(index - 1, ` `                            ``sum, count, arr, K); ` ` `  `    ``// If we pick the current element add it to ` `    ``// our current sum and increment count by 1 ` `    ``int` `pick = waysutil(index - 1, ` `                        ``sum + arr[index], ` `                        ``count + 1, arr, K); ` `    ``int` `total = pick + dontpick; ` ` `  `    ``// Store the value for the current function ` `    ``dp[index][sum][count] = total; ` `    ``return` `total; ` `} ` ` `  `// Function to return the number of ways ` `int` `ways(``int` `N, ``int` `K, ``int``* arr) ` `{ ` `    ``vector<``int``> Arr; ` ` `  `    ``// Push -1 at the beginning to ` `    ``// make it 1-based indexing ` `    ``Arr.push_back(-1); ` `    ``for` `(``int` `i = 0; i < N; ++i) { ` `        ``Arr.push_back(arr[i]); ` `    ``} ` ` `  `    ``// Initialize dp array by -1 ` `    ``memset``(dp, -1, ``sizeof` `dp); ` ` `  `    ``// Call recursive function ` `    ``// waysutil to calculate total ways ` `    ``int` `answer = waysutil(N, 0, 0, Arr, K); ` `    ``return` `answer; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `arr[] = { 3, 6, 2, 8, 7, 6, 5, 9 }; ` `    ``int` `N = ``sizeof``(arr) / ``sizeof``(arr); ` `    ``int` `K = 5; ` `    ``cout << ways(N, K, arr); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java implementation of the above approach ` `import` `java.util.*; ` ` `  `class` `GFG  ` `{ ` ` `  `    ``static` `int` `MAX_INDEX = ``51``; ` `    ``static` `int` `MAX_SUM = ``2505``; ` ` `  `    ``// This dp array is used to store our values ` `    ``// so that we don't have to calculate same ` `    ``// values again and again ` `    ``static` `int``[][][] dp = ``new` `int``[MAX_INDEX][MAX_SUM][MAX_INDEX]; ` ` `  `    ``static` `int` `waysutil(``int` `index, ``int` `sum, ``int` `count, ` `                        ``Vector arr, ``int` `K) ` `    ``{ ` `        ``// Base cases ` `        ``// Index can't be less than 0 ` `        ``if` `(index < ``0``)  ` `        ``{ ` `            ``return` `0``; ` `        ``} ` ` `  `        ``if` `(index == ``0``)  ` `        ``{ ` ` `  `            ``// No element is picked hence ` `            ``// average cannot be calculated ` `            ``if` `(count == ``0``) ` `            ``{ ` `                ``return` `0``; ` `            ``} ` `            ``int` `remainder = sum % count; ` ` `  `            ``// If remainder is non zero, we cannot ` `            ``// divide the sum by count i.e. the average ` `            ``// will not be an integer ` `            ``if` `(remainder != ``0``)  ` `            ``{ ` `                ``return` `0``; ` `            ``} ` `            ``int` `average = sum / count; ` ` `  `            ``// If we find an average return 1 ` `            ``if` `(average == K)  ` `            ``{ ` `                ``return` `1``; ` `            ``} ` `        ``} ` ` `  `        ``// If we have already calculated this function ` `        ``// simply return it instead of calculating it again ` `        ``if` `(dp[index][sum][count] != -``1``)  ` `        ``{ ` `            ``return` `dp[index][sum][count]; ` `        ``} ` ` `  `        ``// If we don't pick the current element ` `        ``// simple recur for index -1 ` `        ``int` `dontpick = waysutil(index - ``1``, ` `                ``sum, count, arr, K); ` ` `  `        ``// If we pick the current element add it to ` `        ``// our current sum and increment count by 1 ` `        ``int` `pick = waysutil(index - ``1``, ` `                ``sum + arr.get(index), ` `                ``count + ``1``, arr, K); ` `        ``int` `total = pick + dontpick; ` ` `  `        ``// Store the value for the current function ` `        ``dp[index][sum][count] = total; ` `        ``return` `total; ` `    ``} ` ` `  `    ``// Function to return the number of ways ` `    ``static` `int` `ways(``int` `N, ``int` `K, ``int``[] arr) ` `    ``{ ` `        ``Vector Arr = ``new` `Vector<>(); ` ` `  `        ``// Push -1 at the beginning to ` `        ``// make it 1-based indexing ` `        ``Arr.add(-``1``); ` `        ``for` `(``int` `i = ``0``; i < N; ++i) ` `        ``{ ` `            ``Arr.add(arr[i]); ` `        ``} ` ` `  `        ``// Initialize dp array by -1 ` `        ``for` `(``int` `i = ``0``; i < MAX_INDEX; i++) ` `        ``{ ` `            ``for` `(``int` `j = ``0``; j < MAX_SUM; j++) ` `            ``{ ` `                ``for` `(``int` `l = ``0``; l < MAX_INDEX; l++)  ` `                ``{ ` `                    ``dp[i][j][l] = -``1``; ` `                ``} ` `            ``} ` `        ``} ` ` `  `        ``// Call recursive function ` `        ``// waysutil to calculate total ways ` `        ``int` `answer = waysutil(N, ``0``, ``0``, Arr, K); ` `        ``return` `answer; ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `main(String args[]) ` `    ``{ ` `        ``int` `arr[] = {``3``, ``6``, ``2``, ``8``, ``7``, ``6``, ``5``, ``9``}; ` `        ``int` `N = arr.length; ` `        ``int` `K = ``5``; ` `        ``System.out.println(ways(N, K, arr)); ` `    ``} ` `} ` ` `  `/* This code contributed by PrinciRaj1992 */`

## Python3

 `# Python implementation of above approach ` `import` `numpy as np ` ` `  `MAX_INDEX ``=` `51` `MAX_SUM ``=` `2505` ` `  `# This dp array is used to store our values  ` `# so that we don't have to calculate same  ` `# values again and again  ` ` `  `# Initialize dp array by -1  ` `dp ``=` `np.ones((MAX_INDEX,MAX_SUM,MAX_INDEX)) ``*` `-``1``;  ` ` `  `def` `waysutil(index, ``sum``, count, arr, K) :  ` ` `  `    ``# Base cases  ` `    ``# Index can't be less than 0  ` `    ``if` `(index < ``0``) : ` `        ``return` `0``;  ` ` `  `    ``if` `(index ``=``=` `0``) : ` ` `  `        ``# No element is picked hence  ` `        ``# average cannot be calculated  ` `        ``if` `(count ``=``=` `0``) : ` `            ``return` `0``; ` `             `  `        ``remainder ``=` `sum` `%` `count;  ` ` `  `        ``# If remainder is non zero, we cannot  ` `        ``# divide the sum by count i.e. the average  ` `        ``# will not be an integer  ` `        ``if` `(remainder !``=` `0``) : ` `            ``return` `0``;  ` `             `  `        ``average ``=` `sum` `/``/` `count;  ` ` `  `        ``# If we find an average return 1  ` `        ``if` `(average ``=``=` `K) : ` `            ``return` `1``;  ` ` `  `    ``# If we have already calculated this function  ` `    ``# simply return it instead of calculating it again  ` `    ``if` `(dp[index][``sum``][count] !``=` `-``1``) : ` `        ``return` `dp[index][``sum``][count];  ` ` `  `    ``# If we don't pick the current element  ` `    ``# simple recur for index -1  ` `    ``dontpick ``=` `waysutil(index ``-` `1``,  ` `                            ``sum``, count, arr, K);  ` ` `  `    ``# If we pick the current element add it to  ` `    ``# our current sum and increment count by 1  ` `    ``pick ``=` `waysutil(index ``-` `1``,  ` `                        ``sum` `+` `arr[index],  ` `                        ``count ``+` `1``, arr, K);  ` `                         `  `    ``total ``=` `pick ``+` `dontpick;  ` ` `  `    ``# Store the value for the current function  ` `    ``dp[index][``sum``][count] ``=` `total;  ` `     `  `    ``return` `total;  ` ` `  ` `  `# Function to return the number of ways  ` `def` `ways(N, K, arr) : ` ` `  `    ``Arr ``=` `[];  ` ` `  `    ``# Push -1 at the beginning to  ` `    ``# make it 1-based indexing  ` `    ``Arr.append(``-``1``);  ` `    ``for` `i ``in` `range``(N) : ` `        ``Arr.append(arr[i]);  ` ` `  `    ``# Call recursive function  ` `    ``# waysutil to calculate total ways  ` `    ``answer ``=` `waysutil(N, ``0``, ``0``, Arr, K);  ` `    ``return` `answer;  ` ` `  ` `  `# Driver code  ` `if` `__name__ ``=``=` `"__main__"` `:  ` ` `  `    ``arr ``=` `[ ``3``, ``6``, ``2``, ``8``, ``7``, ``6``, ``5``, ``9` `];  ` `    ``N ``=``len``(arr);  ` `    ``K ``=` `5``;  ` `    ``print``(ways(N, K, arr));  ` ` `  `# This code is contributed by AnkitRai01 `

## C#

 `// C# implementation of the above approach ` `using` `System; ` `using` `System.Collections.Generic; ` `     `  `class` `GFG  ` `{ ` ` `  `    ``static` `int` `MAX_INDEX = 51; ` `    ``static` `int` `MAX_SUM = 2505; ` ` `  `    ``// This dp array is used to store our values ` `    ``// so that we don't have to calculate same ` `    ``// values again and again ` `    ``static` `int``[,,] dp = ``new` `int``[MAX_INDEX, MAX_SUM, MAX_INDEX]; ` ` `  `    ``static` `int` `waysutil(``int` `index, ``int` `sum, ``int` `count, ` `                        ``List<``int``> arr, ``int` `K) ` `    ``{ ` `        ``// Base cases ` `        ``// Index can't be less than 0 ` `        ``if` `(index < 0)  ` `        ``{ ` `            ``return` `0; ` `        ``} ` ` `  `        ``if` `(index == 0)  ` `        ``{ ` ` `  `            ``// No element is picked hence ` `            ``// average cannot be calculated ` `            ``if` `(count == 0) ` `            ``{ ` `                ``return` `0; ` `            ``} ` `            ``int` `remainder = sum % count; ` ` `  `            ``// If remainder is non zero, we cannot ` `            ``// divide the sum by count i.e. the average ` `            ``// will not be an integer ` `            ``if` `(remainder != 0)  ` `            ``{ ` `                ``return` `0; ` `            ``} ` `            ``int` `average = sum / count; ` ` `  `            ``// If we find an average return 1 ` `            ``if` `(average == K)  ` `            ``{ ` `                ``return` `1; ` `            ``} ` `        ``} ` ` `  `        ``// If we have already calculated this function ` `        ``// simply return it instead of calculating it again ` `        ``if` `(dp[index,sum,count] != -1)  ` `        ``{ ` `            ``return` `dp[index, sum, count]; ` `        ``} ` ` `  `        ``// If we don't pick the current element ` `        ``// simple recur for index -1 ` `        ``int` `dontpick = waysutil(index - 1, ` `                ``sum, count, arr, K); ` ` `  `        ``// If we pick the current element add it to ` `        ``// our current sum and increment count by 1 ` `        ``int` `pick = waysutil(index - 1, ` `                ``sum + arr[index], ` `                ``count + 1, arr, K); ` `        ``int` `total = pick + dontpick; ` ` `  `        ``// Store the value for the current function ` `        ``dp[index,sum,count] = total; ` `        ``return` `total; ` `    ``} ` ` `  `    ``// Function to return the number of ways ` `    ``static` `int` `ways(``int` `N, ``int` `K, ``int``[] arr) ` `    ``{ ` `        ``List<``int``> Arr = ``new` `List<``int``>(); ` ` `  `        ``// Push -1 at the beginning to ` `        ``// make it 1-based indexing ` `        ``Arr.Add(-1); ` `        ``for` `(``int` `i = 0; i < N; ++i) ` `        ``{ ` `            ``Arr.Add(arr[i]); ` `        ``} ` ` `  `        ``// Initialize dp array by -1 ` `        ``for` `(``int` `i = 0; i < MAX_INDEX; i++) ` `        ``{ ` `            ``for` `(``int` `j = 0; j < MAX_SUM; j++) ` `            ``{ ` `                ``for` `(``int` `l = 0; l < MAX_INDEX; l++)  ` `                ``{ ` `                    ``dp[i, j, l] = -1; ` `                ``} ` `            ``} ` `        ``} ` ` `  `        ``// Call recursive function ` `        ``// waysutil to calculate total ways ` `        ``int` `answer = waysutil(N, 0, 0, Arr, K); ` `        ``return` `answer; ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `Main(String []args) ` `    ``{ ` `        ``int` `[]arr = {3, 6, 2, 8, 7, 6, 5, 9}; ` `        ``int` `N = arr.Length; ` `        ``int` `K = 5; ` `        ``Console.WriteLine(ways(N, K, arr)); ` `    ``} ` `} ` ` `  `// This code is contributed by Princi Singh `

Output:

```19
```

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