# 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}

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

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

 `# 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) `

Output:

```144
``` My Personal Notes arrow_drop_up Check out this Author's contributed articles.

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