# Sum of the products of all possible Subsets

Given an array of n non-negative integers. The task is to find the sum of the product of elements of all the possible subsets. It may be assumed that the numbers in subsets are small and computing product doesn’t cause arithmetic overflow.**Example :**

Input : arr[] = {1, 2, 3} Output : 23 Possible Subset are: 1, 2, 3, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} Products of elements in above subsets : 1, 2, 3, 2, 3, 6, 6 Sum of all products = 1 + 2 + 3 + 2 + 3 + 6 + 6 = 23

**Naive Approach: **Simple approach is to generate all possible subset one by one and calculate sum of all elements. Time Complexity of this approach is exponential as there are total 2^{n} – 1 subsets.

An **Efficient approach** is to generalize the whole problem into some pattern. Suppose we have two numbers a and b. We can write all possible subset products as:-

= a + b + ab = a(1+b) + b + 1 - 1 = a(1+b) + (1+b) - 1 = (a + 1) * (b + 1) - 1 = (1+a) * (1 + b) - 1

Now take three numbers a, b, c:-

= a + b + c + ab + bc + ca + abc = a + ac + b + bc + ab + abc + c + 1 - 1 = a(1+c) + b(1+c) + ab(1+c) + c + 1 - 1 = (a + b + ab + 1)(1+c) - 1 = (1+a) * (1+b) * (1+c) - 1

The above pattern can be generalized for n numbers.

Below is the implementation of above idea :

## C++

`// C++ program to find sum of product of` `// all subsets.` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// Returns sum of products of all subsets` `// of arr[0..n-1]` `int` `productOfSubsetSums(` `int` `arr[], ` `int` `n)` `{` ` ` `int` `ans = 1;` ` ` `for` `(` `int` `i = 0; i < n; ++i )` ` ` `ans = ans * (arr[i] + 1);` ` ` `return` `ans-1;` `}` `// Driver code` `int` `main()` `{` ` ` `int` `arr[] = {1, 2, 3, 4};` ` ` `int` `n = ` `sizeof` `(arr)/` `sizeof` `arr[0];` ` ` `cout << productOfSubsetSums(arr, n);` ` ` `return` `0;` `}` |

## Java

`// Java program to find sum of product of` `// all subsets.` `public` `class` `Subset` `{` ` ` `// Returns sum of products of all subsets` ` ` `// of arr[0..n-1]` ` ` `static` `int` `productOfSubsetSums(` `int` `arr[], ` `int` `n)` ` ` `{` ` ` `int` `ans = ` `1` `;` ` ` `for` `(` `int` `i = ` `0` `; i < n; ++i )` ` ` `ans = ans * (arr[i] + ` `1` `);` ` ` `return` `ans-` `1` `;` ` ` `}` ` ` ` ` `public` `static` `void` `main (String[] args)` ` ` `{` ` ` `int` `arr[] = {` `1` `, ` `2` `, ` `3` `, ` `4` `};` ` ` `int` `n = arr.length;` ` ` `System.out.println(productOfSubsetSums(arr, n));` ` ` `}` `}` `// This code is contributed by Saket Kumar` |

## Python3

`# Python3 program to` `# find sum of product of` `# all subsets.` `# Returns sum of products` `# of all subsets` `# of arr[0..n-1]` `def` `productOfSubsetSums(arr, n):` ` ` `ans ` `=` `1` `;` ` ` `for` `i ` `in` `range` `(` `0` `,n):` ` ` `ans ` `=` `ans ` `*` `(arr[i] ` `+` `1` `)` ` ` `return` `ans` `-` `1` `# Driver code` `arr ` `=` `[` `1` `, ` `2` `, ` `3` `, ` `4` `]` `n ` `=` `len` `(arr)` `print` `(productOfSubsetSums(arr, n))` ` ` `# This code is contributed` `# by Shreyanshi Arun.` |

## C#

`// C# program to find sum of` `// product of all subsets.` `using` `System;` `public` `class` `Subset` `{` ` ` ` ` `// Returns sum of products of all` ` ` `// subsets of arr[0..n-1]` ` ` `static` `int` `productOfSubsetSums(` `int` `[]arr, ` `int` `n)` ` ` `{` ` ` `int` `ans = 1;` ` ` `for` `(` `int` `i = 0; i < n; ++i )` ` ` `ans = ans * (arr[i] + 1);` ` ` `return` `ans-1;` ` ` `}` ` ` ` ` `// Driver Code` ` ` `public` `static` `void` `Main ()` ` ` `{` ` ` `int` `[]arr = {1, 2, 3, 4};` ` ` `int` `n = arr.Length;` ` ` `Console.Write(productOfSubsetSums(arr, n));` ` ` `}` `}` `// This code is contributed by Nitin Mittal.` |

## PHP

`<?php` `// PHP program to find sum of` `// product of all subsets.` `// Returns sum of products of` `// all subsets of arr[0..n-1]` `function` `productOfSubsetSums(` `$arr` `, ` `$n` `)` `{` ` ` `$ans` `= 1;` ` ` `for` `(` `$i` `= 0; ` `$i` `< ` `$n` `; ++` `$i` `)` ` ` `$ans` `= ` `$ans` `* (` `$arr` `[` `$i` `] + 1);` ` ` `return` `$ans` `-1;` `}` `// Driver code` `$arr` `= ` `array` `(1, 2, 3, 4);` `$n` `= sizeof(` `$arr` `);` `echo` `(productOfSubsetSums(` `$arr` `, ` `$n` `));` `// This code is contributed by Ajit.` `?>` |

## Javascript

`<script>` `// Javascript program to find sum of product of` `// all subsets.` `// Returns sum of products of all subsets` `// of arr[0..n-1]` `function` `productOfSubsetSums(arr, n)` `{` ` ` `let ans = 1;` ` ` `for` `(let i = 0; i < n; ++i )` ` ` `ans = ans * (arr[i] + 1);` ` ` `return` `ans-1;` `}` `// Driver code` ` ` `let arr = [1, 2, 3, 4];` ` ` `let n = arr.length;` ` ` `document.write(productOfSubsetSums(arr, n));` `// This code is contributed by Mayank Tyagi` `</script>` |

**Output:**

119

**Time Complexity: **O(n) **Auxiliary Space: **O(1)

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