Sum of all subsequences of length K

Given an array arr[]and an integer K, the task is to find the sum of all K length subsequences from the given array.

Example: 

Input: arr[] = {2, 3, 4}, K = 2 
Output: 18 
Explanation: 
There are 3 possible subsequences of length 2 which are {2, 3}, {2, 4} and {3, 4} 
The sum of all 2 length subsequences is 5 + 6 + 7 = 18

Input: arr[] = {7, 8, 9, 2}, K = 2 
Output: 78 
Explanation: 
There are 6 subsequences of length 2 which are {7, 8}, {7, 9}, {7, 2}, {8, 9}, {8, 2} and {9, 2}. 
The sum of all 2 length sub sequences is 15 + 16 + 9 + 17 + 10 + 11 = 78 
 

Approach: 
To solve the problem mentioned above we have to consider all K length subsequence that is "n choose k", i.e. k * nCk 

• The count of total element in all K length subsequences is k * nCk  , possibility of appearing of each element is same.
• So each element appears((k * nCk) / n )  times and it contributes arr[i] * ( (k*nCk)/n )  in the result.
• Hence, the sum of all K length subsequence is sum(array) * ( (k * nCk) / n )

Below is the implementation of the above mentioned approach: 

// C++ implementation to find sum
// of all subsequences of length K

#include <bits/stdc++.h>
using namespace std;

int fact(int n);

// Function to find nCr
int nCr(int n, int r)
{
    return fact(n)
           / (fact(r)
              * fact(n - r));
}

// Function that returns
// factorial of n
int fact(int n)
{
    int res = 1;
    for (int i = 2; i <= n; i++)
        res = res * i;
    return res;
}

// Function for finding sum
// of all K length subsequences
int sumSubsequences(
    int arr[], int n, int k)
{

    int sum = 0;

    // Calculate the sum of array
    for (int i = 0; i < n; i++) {
        sum += arr[i];
    }
    int kLengthSubSequence;

    // Calculate nCk
    kLengthSubSequence = nCr(n, k);

    int ans
        = sum
          * ((k * kLengthSubSequence)
             / n);

    // Return the final result
    return ans;
}

// Driver code
int main()
{

    int arr[] = { 7, 8, 9, 2 };

    int K = 2;

    int n = sizeof(arr) / sizeof(arr[0]);

    cout << sumSubsequences(arr, n, K);
    return 0;
}

// Java implementation to find sum
// of all subsequences of length K
class GFG{

// Function to find nCr
static int nCr(int n, int r)
{
    return fact(n) / (fact(r) * fact(n - r));
}

// Function that returns
// factorial of n
static int fact(int n)
{
    int res = 1;
    for (int i = 2; i <= n; i++)
        res = res * i;
    return res;
}

// Function for finding sum
// of all K length subsequences
static int sumSubsequences(int arr[], 
                           int n, int k)
{
    int sum = 0;

    // Calculate the sum of array
    for (int i = 0; i < n; i++) 
    {
        sum += arr[i];
    }
    int kLengthSubSequence;

    // Calculate nCk
    kLengthSubSequence = nCr(n, k);

    int ans = sum * ((k * kLengthSubSequence) / n);

    // Return the final result
    return ans;
}

// Driver code
public static void main(String[] args)
{
    int arr[] = { 7, 8, 9, 2 };

    int K = 2;

    int n = arr.length;

    System.out.print(sumSubsequences(arr, n, K));
}
}

// This code contributed by Rajput-Ji

# Python3 implementation to find sum 
# of all subsequences of length K

# Function to find nCr 
def nCr(n, r):
    
    return fact(n) / (fact(r) * 
                      fact(n - r))

# Function that returns 
# factorial of n
def fact(n):
    
    res = 1
    for i in range(2, n + 1):
        res = res * i 
    return res
    
# Function for finding sum 
# of all K length subsequences
def sumSubsequences(arr, n, k):
    
    sum = 0
    
    # Calculate the sum of array 
    for i in range(0, n):
        sum = sum + arr[i]
    
    # Calculate nCk     
    kLengthSubSequence = nCr(n, k)
    ans = sum * ((k * kLengthSubSequence) / n);
    
    # Return the final result 
    return ans

# Driver Code 
arr = [ 7, 8, 9, 2 ]
k = 2
n = len(arr)

print(sumSubsequences(arr, n, k))

# This code is contributed by skylags    

// C# implementation to find sum
// of all subsequences of length K
using System;

class GFG{
    
// Function to find nCr
static int nCr(int n, int r)
{
    return fact(n) / (fact(r) * fact(n - r));
}
    
// Function that returns
// factorial of n
static int fact(int n)
{
    int res = 1;
    
    for(int i = 2; i <= n; i++)
       res = res * i;
    return res;
}
    
// Function for finding sum
// of all K length subsequences
static int sumSubsequences(int[] arr, 
                           int n, int k)
{
    int sum = 0;
    
    // Calculate the sum of array
    for(int i = 0; i < n; i++) 
    {
       sum += arr[i];
    }
    
    int kLengthSubSequence;
    
    // Calculate nCk
    kLengthSubSequence = nCr(n, k);
    int ans = sum * ((k * kLengthSubSequence) / n);
    
    // Return the final result
    return ans;
}

// Driver code
static void Main() 
{
    int[] arr = { 7, 8, 9, 2 };
    int K = 2;
    int n = arr.Length;
        
    Console.Write(sumSubsequences(arr, n, K));
}
}

// This code is contributed by divyeshrabadiya07

<script>

// Javascript implementation to find sum
// of all subsequences of length K

// Function to find nCr
function nCr(n, r)
{
    return fact(n) / (fact(r) * 
           fact(n - r));
}

// Function that returns
// factorial of n
function fact(n)
{
    var res = 1;
    for(var i = 2; i <= n; i++)
        res = res * i;
        
    return res;
}

// Function for finding sum
// of all K length subsequences
function sumSubsequences(arr, n, k)
{
    var sum = 0;

    // Calculate the sum of array
    for(var i = 0; i < n; i++)
    {
        sum += arr[i];
    }
    var kLengthSubSequence;

    // Calculate nCk
    kLengthSubSequence = nCr(n, k);

    var ans = sum * ((k * 
              kLengthSubSequence) / n);

    // Return the final result
    return ans;
}

// Driver code
var arr = [ 7, 8, 9, 2 ];
var K = 2;
var n = arr.length;

document.write(sumSubsequences(arr, n, K));

// This code is contributed by noob2000

</script> 

78

Time complexity: O(n),the time complexity of this algorithm is O(n) where n is the length of the array. We use a single loop to iterate over the given array and calculate the sum.
Auxiliary Space complexity: O(1),the space complexity of this algorithm is O(1) as we are not using any extra space.