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

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
Explanation:
Some of the possible permutation of first 5 positive integers, such that prime numbers are at prime indices are: {1, 2, 3, 4}, {1, 3, 2, 4}, {4, 2, 3, 1}, {4, 3, 2, 1}

Approach: There are K number of primes from 1 to N and there is exactly K number of prime indexes from index 1 to N. So the number of permutations for prime numbers is K!. Similarly, the number of permutations for non-prime numbers is (N-K)!. So the total number of permutations is K!*(N-K)!
For example:

Given test case: [1,2,3,4,5]. 
2, 3 and 5 are fixed on prime index slots, 
we can only swap them around. 
There are 3 x 2 x 1 = 3! ways
[[2,3,5], [2,5,3], [3,2,5], 
[3,5,2], [5,2,3], [5,3,2]], 
For Non-prime numbers - {1,4}  
[[1,4], [4,1]]  
So the total is  3!*2!

Below is the implementation of the above approach:

Python

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# Python implementation to find the 
# permutation of first N positive 
# integers such that prime numbers 
# are at the prime indices
  
import math
  
# Function to check that
# a number is prime or not 
def isPrime(n):
    if n <= 1:
        return False
  
    # Loop to check that 
    # number is divisible by any
    # other number or not except 1
    for i in range(2, int(n**0.5)+1):
        if n % i == 0:
            return False
    else:
        return True
          
# Constant value for modulo
CONST = int(math.pow(10, 9))+7
  
# Function to find the permutations
def findPermutations(n):
    prime = 0
      
    # Loop to find the  
    # number of prime numbers
    for j in range (1, n + 1):
        if isPrime(j):
            prime+= 1
      
    # Permutation of N
    # positive integers
    W = math.factorial(prime)*\
      math.factorial(n-prime)
        
    print (W % CONST)
  
# Driver Code
if __name__ == "__main__":
    n = 7
      
    # Function Call
    findPermutations(n)

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Output:

144

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