How do you solve Prime Arrangements on LeetCode?

Prime Arrangements (LeetCode #1175) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n) time complexity and O(1) space complexity. Key insight: Understanding prime numbers and their properties is crucial for this problem.

What is the brute force solution for Prime Arrangements?

The brute force approach for Prime Arrangements is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute force approach involves generating all possible permutations of numbers from 1 to n and checking each permutation to see if prime numbers are placed at prime indices. This is straightforward but inefficient. The optimal Optimal Solution approach improves this to O(n).

What algorithm or data structure is used to solve Prime Arrangements?

Prime Arrangements uses the following concepts: Math. The recommended patterns to study are: Mathematical counting principles, Factorial calculations.

What are the interview tips for Prime Arrangements?

Always clarify the problem constraints and requirements. Think about edge cases, such as the smallest and largest values of n. Practice writing both brute force and optimal solutions to strengthen understanding.

What are common mistakes when solving Prime Arrangements?

Confusing prime indices with prime numbers. Not using modulo when calculating large numbers.

#1175

Prime Arrangements

Easy

MathMathematical counting principlesFactorial calculations

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(n)
Space	O(1)	O(1)

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Intuition

Time O(n)Space O(1)

Instead of generating permutations, we can directly calculate the number of valid arrangements by counting primes and composites. The number of valid arrangements is the product of the factorials of the counts of primes and composites.

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Algorithm

4 steps

1Step 1: Count the number of prime numbers up to n.
2Step 2: Count the number of composite numbers up to n.
3Step 3: Calculate the factorial of the count of primes and the count of composites.
4Step 4: Return the product of these factorials modulo 10^9 + 7.

solution.py10 lines

1def factorial(x):
2    result = 1
3    for i in range(2, x + 1):
4        result = (result * i) % (10**9 + 7)
5    return result
6
7def prime_arrangements(n):
8    primes_count = sum(1 for i in range(2, n + 1) if is_prime(i))
9    composites_count = n - primes_count
10    return (factorial(primes_count) * factorial(composites_count)) % (10**9 + 7)

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Complexity note: The time complexity is O(n) because we only need to count primes and composites, and calculate their factorials, which is efficient compared to generating permutations.

1Understanding prime numbers and their properties is crucial for this problem.
2The factorial function grows rapidly, so we must use modulo to prevent overflow.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.