# Sum of all ordered pair-products from a given array

• Difficulty Level : Easy
• Last Updated : 07 May, 2021

Given an array arr[] of size N, the task is to find the sum of all products of ordered pairs that can be generated from the given array elements.
Examples:

Input: arr[] ={1, 2, 3}
Output: 36
Explanation:All possible pairs are {(1, 1), {1, 2}, {1, 3}, {2, 1}, {2, 2}, {2, 3}, {3, 1}, {3, 2}, {3, 3}}. Therefore, the sum of product of all pairs is 36.

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Input : arr[]={3, 4, 1, 2, 5}
Output: 225

Naive Approach: The simplest approach is to iterate through all possible pairs from the given array calculate the sum of product of all pair-products.

Time Complexity: O(N2)
Auxiliary Space: O(1)

Efficient Approach: The above approach can be optimized based on the following observations:

arr[]={a1, a2, a3, a4 ….., an-1, an}
Now, sum of product of all  possible pairs is =
{
(a1 * a1) + (a1 * a2) + (a1 * a3) + …..+ (a1 * an-1) + (a1, an) +
(a2 * a1) + (a2 * a2) + (a2 * a3) + ….. + (a2 * an-1) + (a2 * an) +
(a3 * a1) + (a3 * a2) + (a3 * a3) + ….. + (a3 * an-1) + (a3, an) +
…………………………………………………………………………..
(an, * a1) + (an * a2), +(an * a3) + ….. + (an * an-1) + (an, an)
}

={
(a1 + a2+  a3 + …..+ an-1 +  an * (a1 + a2+  a3 + …..+ an-1 +  an
}
=(a1 + a2+  a3 + …..+ an-1 +  an )2

Follow the steps below to solve the problem:

1. Initialize the variable, res=0 to store the sum of array elements
2. Traverse the array, arr[] and add each element of the array to res.
3. Finally, print the square of the res as the required answer.

Below is the implementation of the above approach:

## C++

 `// C++ program to implement``// the above approach` `#include ``using` `namespace` `std;` `// Function to calculate the``// sum of all pair-products``int` `sumOfProd(``int` `arr[], ``int` `N)``{``    ``// Stores sum of array``    ``int` `sum = 0;` `    ``for` `(``int` `i = 0; i < N; i++) {` `        ``// Update sum of the array``        ``sum += arr[i];``    ``}` `    ``return` `sum * sum;``}` `// Driver Code``int` `main()``{``    ``int` `arr[] = { 2, 3, 1, 5, 4 };``    ``int` `N = ``sizeof``(arr) / ``sizeof``(arr);``    ``cout << sumOfProd(arr, N);``    ``return` `0;``}`

## Java

 `// Java program to implement``// the above approach``import` `java.util.*;` `class` `GFG{` `// Function to calculate the``// sum of all pair-products``static` `int` `sumOfProd(``int` `arr[], ``int` `N)``{``    ` `    ``// Stores sum of array``    ``int` `sum = ``0``;` `    ``for``(``int` `i = ``0``; i < N; i++)``    ``{``        ` `        ``// Update sum of the array``        ``sum += arr[i];``    ``}``    ``return` `sum * sum;``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``int` `arr[] = { ``2``, ``3``, ``1``, ``5``, ``4` `};``    ``int` `N = arr.length;``    ` `    ``System.out.print(sumOfProd(arr, N));``}``}` `// This code is contributed by offbeat`

## Python3

 `# Python3 program to implement``# the above approach ` `# Function to calculate the``# sum of all pair-products``def` `sumOfProd(arr, N):``    ` `    ``# Stores sum of array``    ``sum` `=` `0` `    ``for` `i ``in` `range``(N):``        ` `        ``# Update sum of the array``        ``sum` `+``=` `arr[i]` `    ``return` `sum` `*` `sum` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ` `    ``arr ``=` `[ ``2``, ``3``, ``1``, ``5``, ``4` `]``    ``N ``=` `len``(arr)``    ` `    ``print``(sumOfProd(arr, N))` `# This code is contributed by SURENDRA_GANGWAR`

## C#

 `// C# program to implement ``// the above approach  ``using` `System;` `class` `GFG{``    ` `// Function to calculate the ``// sum of all pair-products ``static` `int` `sumOfProd(``int``[] arr, ``int` `N) ``{ ``    ` `    ``// Stores sum of array ``    ``int` `sum = 0; ``  ` `    ``for``(``int` `i = 0; i < N; i++)``    ``{``        ` `        ``// Update sum of the array ``        ``sum += arr[i]; ``    ``} ``    ``return` `sum * sum; ``}  ` `// Driver code``static` `void` `Main()``{``    ``int``[] arr = { 2, 3, 1, 5, 4 }; ``    ``int` `N = arr.Length; ``    ` `    ``Console.WriteLine(sumOfProd(arr, N));``}``}` `// This code is contributed by divyeshrabadiya07`

## Javascript

 ``

Output:

`225`

Time Complexity: O(N)

Auxiliary Space: O(1)

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