Leetcode #1175: Prime Arrangements
In this guide, we solve Leetcode #1175 Prime Arrangements in Python and focus on the core idea that makes the solution efficient.
You will see the intuition, the step-by-step method, and a clean Python implementation you can use in interviews.
Problem Statement
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.
Quick Facts
- Difficulty: Easy
- Premium: No
- Tags: Math
Intuition
There is a mathematical invariant or formula that directly leads to the result.
Using math avoids unnecessary loops and reduces complexity.
Approach
Derive the formula or update rule, then compute the answer directly.
Handle edge cases like overflow or zero carefully.
Steps:
- Identify the math relationship.
- Compute the result with a loop or formula.
- Handle edge cases.
Example
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.
Python Solution
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)
Complexity
The time complexity is . The space complexity is O(1).
Edge Cases and Pitfalls
Watch for boundary values, empty inputs, and duplicate values where applicable. If the problem involves ordering or constraints, confirm the invariant is preserved at every step.
Summary
This Python solution focuses on the essential structure of the problem and keeps the implementation interview-friendly while meeting the constraints.