<p>Return the number of permutations of 1 to <code>n</code> so that prime numbers are at prime indices (1-indexed.)</p>

<p><em>(Recall that an integer&nbsp;is prime if and only if it is greater than 1, and cannot be written as a product of two positive integers&nbsp;both smaller than it.)</em></p>

<p>Since the answer may be large, return the answer <strong>modulo <code>10^9 + 7</code></strong>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> n = 5
<strong>Output:</strong> 12
<strong>Explanation:</strong> For example [1,2,5,4,3] is a valid permutation, but [5,2,3,4,1] is not because the prime number 5 is at index 1.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 100
<strong>Output:</strong> 682289015
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 100</code></li>
</ul>

class Solution:
    def numPrimeArrangements(self, n: int) -> int:
        def is_prime(num):
            if num <= 1:
                return False
            if num <= 3:
                return True
            if num % 2 == 0 or num % 3 == 0:
                return False
            i = 5
            while i * i <= num:
                if num % i == 0 or num % (i + 2) == 0:
                    return False
                i += 6
            return True
        
        prime_count = sum(is_prime(i) for i in range(1, n+1))
        non_prime_count = n - prime_count
        
        def factorial(num):
            result = 1
            for i in range(1, num+1):
                result *= i
            return result
        
        MOD = 10**9 + 7
        return (factorial(prime_count) * factorial(non_prime_count)) % MOD
26