# Sum of the sums of all possible subsets

Given an array a of size N. The task is to find the sum of the sums of all possible subsets.

Examples:

Input: a[] = {3, 7}
Output: 20
The subsets are: {3} {7} {3, 7}
{3, 7} = 10
{3} = 3
{7} = 7
10 + 3 + 7 = 20

Input: a[] = {10, 16, 14, 9}
Output: 392

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

Naive Approach: A naive approach is to find all the subsets using power set and then summate all the possible subsets to get the answer.

Time Complexity: O(2N)

Efficient Approach: An efficient approach is to solve the problem using observation. If we write all the subsequences, a common point of observation is that each number appears 2(N – 1) times in a subset and hence will lead to the 2(N-1) as the contribution to the sum. Iterate through the array and add (arr[i] * 2N-1) to the answer.

Below is the implementation of the above approach:

## C++

 `// C++ program to find the sum of ` `// the addition of all possible subsets. ` `#include ` `using` `namespace` `std; ` ` `  `// Function to find the sum ` `// of sum of all the subset ` `int` `sumOfSubset(``int` `a[], ``int` `n) ` `{ ` `    ``int` `times = ``pow``(2, n - 1); ` ` `  `    ``int` `sum = 0; ` ` `  `    ``for` `(``int` `i = 0; i < n; i++) { ` `        ``sum = sum + (a[i] * times); ` `    ``} ` ` `  `    ``return` `sum; ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` `    ``int` `a[] = { 3, 7 }; ` `    ``int` `n = ``sizeof``(a) / ``sizeof``(a); ` `    ``cout << sumOfSubset(a, n); ` `} `

## Java

 `// Java program to find the sum of ` `// the addition of all possible subsets. ` `class` `GFG ` `{ ` `     `  `// Function to find the sum ` `// of sum of all the subset ` `static` `int` `sumOfSubset(``int` `[]a, ``int` `n) ` `{ ` `    ``int` `times = (``int``)Math.pow(``2``, n - ``1``); ` ` `  `    ``int` `sum = ``0``; ` ` `  `    ``for` `(``int` `i = ``0``; i < n; i++)  ` `    ``{ ` `        ``sum = sum + (a[i] * times); ` `    ``} ` ` `  `    ``return` `sum; ` `} ` ` `  `// Driver Code ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``int` `[]a = { ``3``, ``7` `}; ` `    ``int` `n = a.length; ` `    ``System.out.println(sumOfSubset(a, n)); ` `} ` `} ` ` `  `// This code is contributed by 29AjayKumar `

## Python3

 `# Python3 program to find the Sum of ` `# the addition of all possible subsets. ` ` `  `# Function to find the sum ` `# of sum of all the subset ` `def` `SumOfSubset(a, n): ` ` `  `    ``times ``=` `pow``(``2``, n ``-` `1``) ` ` `  `    ``Sum` `=` `0` ` `  `    ``for` `i ``in` `range``(n): ` `        ``Sum` `=` `Sum` `+` `(a[i] ``*` `times) ` ` `  `    ``return` `Sum` ` `  `# Driver Code ` `a ``=` `[``3``, ``7``] ` `n ``=` `len``(a) ` `print``(SumOfSubset(a, n)) ` ` `  `# This code is contributed by Mohit Kumar `

## C#

 `// C# program to find the sum of ` `// the addition of all possible subsets. ` `using` `System; ` ` `  `class` `GFG ` `{ ` `     `  `// Function to find the sum ` `// of sum of all the subset ` `static` `int` `sumOfSubset(``int` `[]a, ``int` `n) ` `{ ` `    ``int` `times = (``int``)Math.Pow(2, n - 1); ` ` `  `    ``int` `sum = 0; ` ` `  `    ``for` `(``int` `i = 0; i < n; i++)  ` `    ``{ ` `        ``sum = sum + (a[i] * times); ` `    ``} ` ` `  `    ``return` `sum; ` `} ` ` `  `// Driver Code ` `public` `static` `void` `Main() ` `{ ` `    ``int` `[]a = { 3, 7 }; ` `    ``int` `n = a.Length; ` `    ``Console.Write(sumOfSubset(a, n)); ` `} ` `} ` ` `  `// This code is contributed by Nidhi `

Output:

```20
```

Time Complexity: O(N)
Space Complexity: O(1)

Note: If N is large, the answer can overflow, thereby use larger data-type.

My Personal Notes arrow_drop_up Striver(underscore)79 at Codechef and codeforces D

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