Given an integer K and an array arr[], the task is to find the number of subsequences from the given array such that each subsequence consists exactly K prime numbers.
Example:
Input: K = 2, arr = [2, 3, 4, 6]
Output: 4
Explanation:
There are 4 subsequences which consists exactly 2 prime numbers {2, 3} {2, 3, 4} {2, 3, 6} {2, 3, 4, 6}
Input: K = 3, arr = [1, 2, 3, 4, 5, 6, 7]
Output: 32
Explanation:
There are 32 subsequences which consists exactly 3 prime numbers.
Approach: To solve the problem mentioned above we have to find the count of prime numbers in the given array.
- Let the count of prime numbers is m.
- We can choose any K integers among m prime numbers.
- So the possible combination of choosing prime numbers in the subsequence is
and in the subsequence we can add any number of non – prime numbers because there is no restriction on the non prime numbers where count of non prime numbers will be (N – m). - We can choose any subset of the (N – m) for nonprime numbers in our subsequence.
- Possibility of choosing all subset of size (N – m) is pow(2, (m – N)).
- For generating subsequences, we will multiply prime number possibility with nonprime number possibility.
and in the subsequence we can add any number of non – prime numbers because there is no restriction on the non prime numbers where count of non prime numbers will be (N – m).Sub sequence count = (Count of prime numbers) C (K) * pow(2, count of non-prime numbers)
Below is the implementation of the above approach:
C++
// C++ implementation to find// the count of subsequences// which consist exactly K primes#include <bits/stdc++.h>using namespace std;// Returns factorial of nint fact(int n){ int res = 1; for (int i = 2; i <= n; i++) res = res * i; return res;}// Function to return total// number of combinationsint nCr(int n, int r){ return fact(n) / (fact(r) * fact(n - r));}// Function check whether a number// is prime or notbool isPrime(int n){ // Corner case if (n <= 1) return false; // Check from 2 to n-1 for (int i = 2; i < n; i++) if (n % i == 0) return false; return true;}// Function for finding number of subsequences// which consists exactly K primesint countSubsequences(int arr[], int n, int k){ int countPrime = 0; for (int i = 0; i < n; i++) { if (isPrime(arr[i])) countPrime++; } // if number of primes are less than k if (countPrime < k) return 0; return nCr(countPrime, k) * pow(2, (n - countPrime));}// Driver codeint main(){ int arr[] = { 1, 2, 3, 4, 5, 6, 7 }; int K = 3; int n = sizeof(arr) / sizeof(arr[0]); cout << countSubsequences(arr, n, K); return 0;} |
Java
// Java implementation to find the// count of subsequences which // consist exactly K primesimport java.util.*;class GFG{ // Returns factorial of nstatic int fact(int n){ int res = 1; for(int i = 2; i <= n; i++) res = res * i; return res;}// Function to return total// number of combinationsstatic int nCr(int n, int r){ return fact(n) / (fact(r) * fact(n - r));}// Function check whether a number// is prime or notstatic boolean isPrime(int n){ // Corner case if (n <= 1) return false; // Check from 2 to n-1 for(int i = 2; i < n; i++) if (n % i == 0) return false; return true;}// Function for finding number of subsequences// which consists exactly K primesstatic int countSubsequences(int arr[], int n, int k){ int countPrime = 0; for(int i = 0; i < n; i++) { if (isPrime(arr[i])) countPrime++; } // If number of primes are less than k if (countPrime < k) return 0; return nCr(countPrime, k) * (int)Math.pow(2, (n - countPrime));}// Driver codepublic static void main(String args[]){ int arr[] = { 1, 2, 3, 4, 5, 6, 7 }; int K = 3; int n = arr.length; System.out.println(countSubsequences(arr, n, K));}}// This code is contributed by ANKITKUMAR34 |
Python3
# Python3 implementation to find the# count of subsequences which consist# exactly K primes# Returns factorial of ndef fact(n): res = 1; for i in range(2, n + 1): res = res * i return res# Function to return total# number of combinationsdef nCr(n, r): return (fact(n) // (fact(r) * fact(n - r)))# Function check whether a number# is prime or notdef isPrime(n): # Corner case if (n <= 1): return False; # Check from 2 to n-1 for i in range(2, n): if (n % i == 0): return False return True# Function for finding number of subsequences# which consists exactly K primesdef countSubsequences(arr, n, k): countPrime = 0 for i in range(n): if (isPrime(arr[i])): countPrime += 1 # If number of primes are less than k if (countPrime < k): return 0 return (nCr(countPrime, k) * pow(2, (n - countPrime)))# Driver codearr = [ 1, 2, 3, 4, 5, 6, 7 ]K = 3n = len(arr)print(countSubsequences(arr, n, K))# This code is contributed by ANKITKUMAR34 |
C#
// C# implementation to find the// count of subsequences which // consist exactly K primesusing System;class GFG{ // Returns factorial of nstatic int fact(int n){ int res = 1; for(int i = 2; i <= n; i++) res = res * i; return res;}// Function to return total// number of combinationsstatic int nCr(int n, int r){ return fact(n) / (fact(r) * fact(n - r));}// Function check whether a number// is prime or notstatic bool isPrime(int n){ // Corner case if (n <= 1) return false; // Check from 2 to n-1 for(int i = 2; i < n; i++) if (n % i == 0) return false; return true;}// Function for finding number of subsequences// which consists exactly K primesstatic int countSubsequences(int []arr, int n, int k){ int countPrime = 0; for(int i = 0; i < n; i++) { if (isPrime(arr[i])) countPrime++; } // If number of primes are less than k if (countPrime < k) return 0; return nCr(countPrime, k) * (int)Math.Pow(2, (n - countPrime));}// Driver codepublic static void Main(String []args){ int []arr = { 1, 2, 3, 4, 5, 6, 7 }; int K = 3; int n = arr.Length; Console.WriteLine(countSubsequences(arr, n, K));}}// This code is contributed by gauravrajput1 |
Javascript
<script>// Javascript implementation to find// the count of subsequences// which consist exactly K primes// Returns factorial of nfunction fact(n){ var res = 1; for (var i = 2; i <= n; i++) res = res * i; return res;}// Function to return total// number of combinationsfunction nCr(n, r){ return fact(n) / (fact(r) * fact(n - r));}// Function check whether a number// is prime or notfunction isPrime(n){ // Corner case if (n <= 1) return false; // Check from 2 to n-1 for (var i = 2; i < n; i++) if (n % i == 0) return false; return true;}// Function for finding number of subsequences// which consists exactly K primesfunction countSubsequences(arr, n, k){ var countPrime = 0; for (var i = 0; i < n; i++) { if (isPrime(arr[i])) countPrime++; } // if number of primes are less than k if (countPrime < k) return 0; return nCr(countPrime, k) * Math.pow(2, (n - countPrime));}// Driver codevar arr = [ 1, 2, 3, 4, 5, 6, 7 ];var K = 3;var n = arr.length;document.write( countSubsequences(arr, n, K));</script> |
Output:
32
Time Complexity: O(N^2) .
Auxiliary Space: O(1) .
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