Sum of all subsequences of length K

Difficulty Level : Hard
Last Updated : 02 Jul, 2020

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

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:

To solve the problem mentioned above we have to consider all K length subsequence that is “n choose k”, i.e. $nCk$ .

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++

// 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

// 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

# 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#

// 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 

Output:

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.

My Personal Notes

Related Articles

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

Python3

C#