Sum of all subsets of a set formed by first n natural numbers

Given a number n, we need to find the sum of all the elements from all possible subsets of a set formed by first n natural numbers.

Examples :

Input :  n = 2
Output : 6
Possible subsets are {{1}, {2}, 
{1, 2}}. Sum of elements in subsets
is 1 + 2 + 1 + 2 = 6

Input :  n = 3
Output : 24
Possible subsets are {{1}, {2}, {3}, 
{1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
Sum of subsets is : 
1 + 2 + 3 + (1 + 2) + (1 + 3) + 
(2 + 3) + (1 + 2 + 3)



A simple solution is to generate all subsets. For every subset, compute its sum and finally return overall sum.

An efficient solution is based on the fact that every number from 1 to n appears exactly 2(n-1) times. So our required sum is (1 + 2 + 3 + ..+ n) * 2(n-1). The sum can be written as (n * (n + 1)/2) * 2(n-1)

C++

// CPP program to find sum of all subsets
// of a set.
#include <bits/stdc++.h>
using namespace std;
  
unsigned long long findSumSubsets(int n)
{
    // sum of subsets is (n * (n + 1) / 2) *
    // pow(2, n-1)
    return (n * (n + 1) / 2) * (1 << (n - 1));
}
  
int main()
{
    int n = 3;
    cout << findSumSubsets(n);
    return 0;
}

Java

// Java program to find sum of all subsets
// of a set.
  
class GFG {
    static long findSumSubsets(int n)
    {
        // sum of subsets is (n * (n + 1) / 2) *
        // pow(2, n-1)
        return (n * (n + 1) / 2) * (1 << (n - 1));
    }
  
    // Driver code
    public static void main(String[] args)
    {
        int n = 3;
        System.out.print(findSumSubsets(n));
    }
}
  
// This code is contributed by Anant Agarwal.

Python3

# Python program to find
# sum of all subsets
# of a set.
  
def findSumSubsets( n):
  
    # sum of subsets 
    # is (n * (n + 1) / 2) *
    # pow(2, n-1)
    return (n * (n + 1) / 2) * (1 << (n - 1))
      
# Driver code     
n = 3
print(findSumSubsets(n))
  
# This code is contributed 
# by sunnysingh.

C#

// C# program to find sum of all subsets
// of a set.
using System;
  
class GFG {
  
    static long findSumSubsets(int n)
    {
  
        // sum of subsets is (n * (n + 1) / 2) *
        // pow(2, n-1)
        return (n * (n + 1) / 2) * (1 << (n - 1));
    }
  
    // Driver code
    public static void Main()
    {
        int n = 3;
  
        Console.WriteLine(findSumSubsets(n));
    }
}
  
// This code is contributed by vt_m.

PHP

<?php
// PHP program to find sum 
// of all subsets of a set
  
function findSumSubsets($n)
{
    // sum of subsets is (n *
    // (n + 1) / 2) * pow(2, n-1)
    return ($n * ($n + 1) / 2) * 
                 (1 << ($n - 1));
}
  
// Driver Code
$n = 3;
echo findSumSubsets($n);
  
// This code is contributed by ajit
?>

Output :

24

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Improved By : vt_m, jit_t




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