#1175

Easy

auto_awesomeMath-driven solution strategy

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Prime Arrangements

Calculate the number of valid permutations of 1 to n where prime numbers occupy prime indices using math-driven logic.

Problem Statement

Given an integer n, return the total number of permutations of the numbers 1 through n such that all prime numbers appear at prime indices, using 1-based indexing. An integer is prime if it is greater than 1 and has no divisors other than 1 and itself. The result should be returned modulo 10^9 + 7 to handle large outputs.

For example, if n = 5, valid permutations include [1,2,5,4,3] because primes 2,3,5 occupy prime positions, but [5,2,3,4,1] is invalid since prime 5 is at index 1. The problem emphasizes a math-driven solution strategy that calculates arrangements separately for primes and non-primes instead of enumerating all permutations.

Examples

Example 1

Input: n = 5

Output: 12

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

Example details omitted.

Constraints

1 <= n <= 100

Solution Approach

Count Primes and Non-Primes

Use a prime-sieving method or direct prime check to count how many numbers between 1 and n are prime. The remaining numbers are non-prime. This separation is crucial for computing arrangements at valid positions and avoids placing primes incorrectly.

Compute Factorials Modulo 10^9 + 7

Calculate factorials for the number of primes and non-primes separately, applying modulo 10^9 + 7 at each multiplication step. This ensures large numbers do not overflow and directly provides the counts of permutations for each group.

Multiply Prime and Non-Prime Arrangements

Multiply the factorial of the prime count by the factorial of the non-prime count modulo 10^9 + 7. This gives the total number of valid prime arrangements and directly addresses the failure mode of trying to brute-force all permutations, which would be inefficient.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity is O(n log log n) using a sieve to count primes, plus O(n) for factorial computations. Space complexity is O(n) for the sieve and factorial storage. The dominant factor is prime counting and factorial computation.

What Interviewers Usually Probe

Check if the candidate separates prime and composite numbers correctly.
Observe whether modulo operations are applied to avoid overflow.
Listen for explanations about factorial usage and arrangement counting.

Common Pitfalls or Variants

Common pitfalls

Attempting to generate all permutations, which is computationally infeasible.
Forgetting to apply modulo 10^9 + 7 during factorial multiplication.
Misidentifying prime numbers, especially 1, which is not prime.

Follow-up variants

Compute valid arrangements where primes occupy only even indices instead of prime indices.
Find permutations for a given list of numbers instead of 1 to n following the prime arrangement pattern.
Return the lexicographically smallest valid permutation instead of counting all permutations.

FAQ

What is the main pattern in the Prime Arrangements problem?

The key pattern is arranging prime numbers at prime indices and non-primes at remaining positions, separating factorial counts for each group.

Why do we return the answer modulo 10^9 + 7?

Because the number of permutations grows quickly, modulo 10^9 + 7 keeps results within integer limits and prevents overflow.

How do I count primes efficiently for this problem?

Use a sieve of Eratosthenes or a direct primality check up to n to identify prime numbers without checking every permutation.

Can this approach handle n = 100?

Yes, counting primes and computing factorials modulo 10^9 + 7 scales efficiently up to n = 100, avoiding brute-force computation.

What are common mistakes to avoid in Prime Arrangements?

Common mistakes include misclassifying 1 as prime, neglecting modulo operations, and attempting full permutation generation instead of factorial multiplication.

terminal

Solution

Solution 1: Mathematics

First, count the number of prime numbers within the range $[1,n]$, which we denote as $cnt$. Then, calculate the product of the factorial of $cnt$ and $n-cnt$ to get the answer, remember to perform the modulo operation.

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class Solution:
    def numPrimeArrangements(self, n: int) -> int:
        def count(n):
            cnt = 0
            primes = [True] * (n + 1)
            for i in range(2, n + 1):
                if primes[i]:
                    cnt += 1
                    for j in range(i + i, n + 1, i):
                        primes[j] = False
            return cnt

        cnt = count(n)
        ans = factorial(cnt) * factorial(n - cnt)
        return ans % (10**9 + 7)

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