Number of ways to calculate a target number using only array elements

Given an integer array, find number of ways to calculate a target number using only array elements and addition or subtraction operator.

Example:

Input: arr[] = {-3, 1, 3, 5}, k = 6
Output: 4
Explanation - 
- (-3) + (3)
+ (1) + (5)
+ (-3) + (1) + (3) + (5)
- (-3) + (1) - (3) + (5)

Input: arr[] = {2, 3, -4, 4}, k = 5
Output: 6
Explanation - 
+ (2) + (3)
+ (2) + (3) + (4) + (-4)
+ (2) + (3) - (4) - (-4)
- (3) + (4) - (-4)
- (2) + (3) + (4)
- (2) + (3) - (-4)

The problem is similar to 0-1 Knapsack Problem where for every item, we either pick the complete item, or don’t pick it at all (0-1 property). The idea remains the same here i.e. we either include the current digit or ignore it. If we include the current digit, we subtract or add it from remaining target and recurse for remaining digits with new target. If target reaches 0, we increment the count. If we have processed all elements of the array and target is not reached, count remains unchanged.

Below is recursive implementation of above idea.  

C++

// C++ program to find the number of ways to calculate
// a target number using only array elements and
// addition or subtraction operator.
#include <iostream>
#include <vector>
using namespace std;

// Function to find the number of ways to calculate
// a target number using only array elements and
// addition or subtraction operator.
int findTotalWays(vector<int> arr, int i, int k)
{

    // If target is reached, return 1
    if (k == 0 && i == arr.size())
        return 1;

    // If all elements are processed and
    // target is not reached, return 0
    if (i >= arr.size())
        return 0;

    // Return total count of three cases
    // 1. Don't consider current element
    // 2. Consider current element and subtract it from target
    // 3. Consider current element and add it to target
    return findTotalWays(arr, i + 1, k)
           + findTotalWays(arr, i + 1, k - arr[i])
           + findTotalWays(arr, i + 1, k + arr[i]);
}

// Driver Program
int main()
{
    vector<int> arr = { -3, 1, 3, 5, 7 };

    // target number
    int k = 6;

    cout << findTotalWays(arr, 0, k) << endl;

    return 0;
}

Java

// Java program to find the number
// of ways to calculate a target
// number using only array elements and
// addition or subtraction operator.
import java.util.*;

class GFG {

    // Function to find the number of ways to calculate
    // a target number using only array elements and
    // addition or subtraction operator.
    static int findTotalWays(Vector<Integer> arr, int i, int k)
    {

        // If target is reached, return 1
        if (k == 0 && i == arr.size()) {
            return 1;
        }

        // If all elements are processed and
        // target is not reached, return 0
        if (i >= arr.size()) {
            return 0;
        }

        // Return total count of three cases
        // 1. Don't consider current element
        // 2. Consider current element and subtract it from target
        // 3. Consider current element and add it to target
        return findTotalWays(arr, i + 1, k)
            + findTotalWays(arr, i + 1, k - arr.get(i))
            + findTotalWays(arr, i + 1, k + arr.get(i));
    }

    // Driver code
    public static void main(String[] args)
    {
        int[] arr = { -3, 1, 3, 5 };
        Vector<Integer> v = new Vector<Integer>();
        for (int a : arr) {
            v.add(a);
        }

        // target number
        int k = 6;

        System.out.println(findTotalWays(v, 0, k));
    }
}

// This code contributed by Rajput-Ji

Python3

# Python 3 program to find the number of
# ways to calculate a target number using
# only array elements and addition or
# subtraction operator.

# Function to find the number of ways to
# calculate a target number using only
# array elements and addition or
# subtraction operator.
def findTotalWays(arr, i, k):

    # If target is reached, return 1
    if (k == 0 and i == len(arr)):
        return 1

    # If all elements are processed and
    # target is not reached, return 0
    if (i >= len(arr)):
        return 0

    # Return total count of three cases
    # 1. Don't consider current element
    # 2. Consider current element and
    # subtract it from target
    # 3. Consider current element and
    # add it to target
    return (findTotalWays(arr, i + 1, k) +
            findTotalWays(arr, i + 1, k - arr[i]) +
            findTotalWays(arr, i + 1, k + arr[i]))

# Driver Code
if __name__ == '__main__':
    arr = [-3, 1, 3, 5, 7]

    # target number
    k = 6

    print(findTotalWays(arr, 0, k))

# This code is contributed by
# Surendra_Gangwar

C#

// C# program to find the number
// of ways to calculate a target
// number using only array elements and
// addition or subtraction operator.
using System;
using System.Collections.Generic;

class GFG {

    // Function to find the number of ways to calculate
    // a target number using only array elements and
    // addition or subtraction operator.
    static int findTotalWays(List<int> arr, int i, int k)
    {
        // If target is reached, return 1
        if (k == 0 && i ==  arr.Count) {
            return 1;
        }

        // If all elements are processed and
        // target is not reached, return 0
        if (i >= arr.Count) {
            return 0;
        }

        // Return total count of three cases
        // 1. Don't consider current element
        // 2. Consider current element and subtract it from target
        // 3. Consider current element and add it to target
        return findTotalWays(arr, i + 1, k)
            + findTotalWays(arr, i + 1, k - arr[i])
            + findTotalWays(arr, i + 1, k + arr[i]);
    }

    // Driver code
    public static void Main(String[] args)
    {
        int[] arr = { -3, 1, 3, 5, 7 };
        List<int> v = new List<int>();
        foreach(int a in arr)
        {
            v.Add(a);
        }

        // target number
        int k = 6;

        Console.WriteLine(findTotalWays(v, 0, k));
    }
}

// This code has been contributed by 29AjayKumar

Javascript

<script>

// Javascript program to find the number
// of ways to calculate a target
// number using only array elements and
// addition or subtraction operator.

// Function to find the number of ways to
// calculate a target number using only
// array elements and addition or
// subtraction operator.
function findTotalWays(arr, i, k)
{

    // If target is reached, return 1
    if (k == 0 && i == arr.length)
    {
        return 1;
    }

    // If all elements are processed and
    // target is not reached, return 0
    if (i >= arr.length)
    {
        return 0;
    }

    // Return total count of three cases
    // 1. Don't consider current element
    // 2. Consider current element and
    //    subtract it from target
    // 3. Consider current element and
    //    add it to target
    return findTotalWays(arr, i + 1, k) +
           findTotalWays(arr, i + 1, k - arr[i]) +
           findTotalWays(arr, i + 1, k + arr[i]);
}

// Driver code
let arr = [ -3, 1, 3, 5 ,7 ];
let k = 6;

document.write(findTotalWays(arr, 0, k));

// This code is contributed by rag2127

</script>

Output :

10

Time Complexity: O(3^n)
Auxiliary Space: O(n) 

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