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1175. Prime Arrangements

Difficulty Topics

Description

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

(Recall that an integer is prime if and only if it is greater than 1, and cannot be written as a product of two positive integers both smaller than it.)

Since the answer may be large, return the answer modulo 10^9 + 7.

 

Example 1:

Input: n = 5
Output: 12
Explanation: 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.

Example 2:

Input: n = 100
Output: 682289015

 

Constraints:

  • 1 <= n <= 100

Solution

prime-arrangements.py
class Solution:
    def numPrimeArrangements(self, n: int) -> int:
        primes = [True] * (n + 1)

        for prime in range(2, int(math.sqrt(n)) + 1):
            if primes[prime]:
                for composite in range(prime * prime, n + 1, prime):
                    primes[composite] = False

        cnt = sum(primes[2:])

        return math.factorial(cnt) * math.factorial(n - cnt) % (10**9 + 7)