# 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 < size; ++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; ` `} ` |

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## 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 ` |

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## 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. ` |

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## 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. ` |

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## 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. ` `?> ` |

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**Output:**

119

**Time Complexity: ** O(n)

**Auxiliary Space: ** O(1)

This article is contributed by Shubham Bansal. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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