Permutation of first N positive integers such that prime numbers are at prime indices | Set 2

Given an integer N, the task is to find the number of permutations of first N positive integers such that prime numbers are at prime indices (for 1-based indexing).
Note: Since, the number of ways may be very large, return the answer modulo 10⁹ + 7.
Examples:

Input: N = 3 
Output: 2 
Explanation:
Possible permutation of first 3 positive integers, such that prime numbers are at prime indices are: {1, 2, 3}, {1, 3, 2}
Input: N = 5 
Output: 12 

Approach: Using Sieve of Eratosthenes

• First, count all the primes between 1 to N using Sieve of Eratosthenes.
• Next, iterate over each position and get the count of prime positions, call it k.
• So, for the k prime numbers, we have limited choice, we need to arrange them in k prime spots.
• For the n-k non-prime numbers, we also have limited choice. We need to arrange them in n-k non-prime spots.
• Both the events are independent, so the total ways would be the product of them.
• Number of ways to arrange k objects in k boxes is k!

Below is the implementation of the above approach: 

C++

// C++ program to count
// permutations from 1 to N
// such that prime numbers
// occur at prime indices

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

static const int MOD = 1e9 + 7;

int numPrimeArrangements(int n)
{
    vector<bool> prime(n + 1, true);
    prime[0] = false;
    prime[1] = false;

    // Computing count of prime
    // numbers using sieve
    for (int i = 2; i <= sqrt(n); i++) {
        if (prime[i])
            for (int factor = 2;
                 factor * i <= n;
                 factor++)
                prime[factor * i] = false;
    }

    int primeIndices = 0;
    for (int i = 1; i <= n; i++)
        if (prime[i])
            primeIndices++;

    int mod = 1e9 + 7, res = 1;

    // Computing permutations for primes
    for (int i = 1; i <= primeIndices; i++)
        res = (1LL * res * i) % mod;

    // Computing permutations for non-primes
    for (int i = 1; i <= (n - primeIndices); i++)
        res = (1LL * res * i) % mod;

    return res;
}

// Driver program
int main()
{
    int N = 5;
    cout << numPrimeArrangements(N);
    return 0;
}

Java

// Java program to count
// permutations from 1 to N
// such that prime numbers
// occur at prime indices


import java.util.*;

class GFG{

static int MOD = (int) (1e9 + 7);

static int numPrimeArrangements(int n)
{
    boolean []prime = new boolean[n + 1];
    Arrays.fill(prime, true);
    prime[0] = false;
    prime[1] = false;

    // Computing count of prime
    // numbers using sieve
    for (int i = 2; i <= Math.sqrt(n); i++) {
        if (prime[i])
            for (int factor = 2;
                 factor * i <= n;
                 factor++)
                prime[factor * i] = false;
    }

    int primeIndices = 0;
    for (int i = 1; i <= n; i++)
        if (prime[i])
            primeIndices++;

    int mod = (int) (1e9 + 7), res = 1;

    // Computing permutations for primes
    for (int i = 1; i <= primeIndices; i++)
        res = (int) ((1L * res * i) % mod);

    // Computing permutations for non-primes
    for (int i = 1; i <= (n - primeIndices); i++)
        res = (int) ((1L * res * i) % mod);

    return res;
}

// Driver program
public static void main(String[] args)
{
    int N = 5;
    System.out.print(numPrimeArrangements(N));
}
}

// This code contributed by sapnasingh4991

Python3

# Python3 program to count
# permutations from 1 to N
# such that prime numbers
# occur at prime indices
import math;

def numPrimeArrangements(n):

    prime = [True for i in range(n + 1)]

    prime[0] = False
    prime[1] = False

    # Computing count of prime
    # numbers using sieve
    for i in range(2,int(math.sqrt(n)) + 1):
        if prime[i]:
            factor = 2

            while factor * i <= n:
                prime[factor * i] = False
                factor += 1

    primeIndices = 0
    for i in range(1, n + 1):
        if prime[i]:
            primeIndices += 1

    mod = 1000000007
    res = 1

    # Computing permutations for primes
    for i in range(1, primeIndices + 1):
        res = (res * i) % mod

    # Computing permutations for non-primes
    for i in range(1, n - primeIndices + 1):
        res = (res * i) % mod

    return res

# Driver code
if __name__=='__main__':

    N = 5

    print(numPrimeArrangements(N))

# This code is contributed by rutvik_56

C#

// C# program to count permutations
// from 1 to N such that prime numbers
// occur at prime indices
using System;

class GFG{

//static int MOD = (int) (1e9 + 7);

static int numPrimeArrangements(int n)
{
    bool []prime = new bool[n + 1];

    for(int i = 0; i < prime.Length; i++)
       prime[i] = true;
    prime[0] = false;
    prime[1] = false;

    // Computing count of prime
    // numbers using sieve
    for(int i = 2; i <= Math.Sqrt(n); i++)
    {
       if (prime[i])
       {
           for(int factor = 2;
                   factor * i <= n;
                   factor++)
              prime[factor * i] = false;
       }
    }

    int primeIndices = 0;
    for(int i = 1; i <= n; i++)
       if (prime[i])
           primeIndices++;

    int mod = (int) (1e9 + 7), res = 1;

    // Computing permutations for primes
    for(int i = 1; i <= primeIndices; i++)
       res = (int) ((1L * res * i) % mod);

    // Computing permutations for non-primes
    for(int i = 1; i <= (n - primeIndices); i++)
       res = (int) ((1L * res * i) % mod);

    return res;
}

// Driver code
public static void Main(String[] args)
{
    int N = 5;

    Console.Write(numPrimeArrangements(N));
}
}

// This code is contributed by gauravrajput1

Javascript

<script>

// javascript program to count
// permutations from 1 to N
// such that prime numbers
// occur at prime indices
var MOD = parseInt(1e9 + 7);

function numPrimeArrangements(n)
{
    var prime = Array.from({length: n+1}, (_, i) => true);
    prime[0] = false;
    prime[1] = false;

    // Computing count of prime
    // numbers using sieve
    for (var i = 2; i <= Math.sqrt(n); i++) {
        if (prime[i])
            for (factor = 2;
                 factor * i <= n;
                 factor++)
                prime[factor * i] = false;
    }

    var primeIndices = 0;
    for (var i = 1; i <= n; i++)
        if (prime[i])
            primeIndices++;

    var mod = parseInt( (1e9 + 7)), res = 1;

    // Computing permutations for primes
    for (var i = 1; i <= primeIndices; i++)
        res = ((1 * res * i) % mod);

    // Computing permutations for non-primes
    for (var i = 1; i <= (n - primeIndices); i++)
        res = ((1 * res * i) % mod);

    return res;
}

// Driver program
var N = 5;
document.write(numPrimeArrangements(N));

// This code contributed by shikhasingrajput
</script>

Output:

12

Time Complexity: O(N * log(log(N)))

Auxiliary Space: O(N)