**π‘ Problem Formulation:** The task at hand is to determine the number of ways to place `n`

rooks on an `n x n`

chessboard such that none of the rooks can attack each other. In other words, we require that no two rooks share the same row or column. Given an integer `n`

, the desired output is a single integer representing the count of distinct valid arrangements.

## Method 1: Factorial Calculation

This method leverages the mathematical fact that the number of ways to arrange `n`

rooks is the same as the number of permutations of `n`

items, which is the factorial of `n`

(written as `n!`

). We use Python’s built-in library `math`

to find the factorial.

Here’s an example:

import math def number_of_ways(n): return math.factorial(n) # Example usage: arrangements = number_of_ways(4) print("Number of arrangements for 4 rooks:", arrangements)

Output: Number of arrangements for 4 rooks: 24

By calling `math.factorial(n)`

, this code snippet calculates the number of ways to uniquely arrange `n`

rooks on a chessboard. It is both efficient and concise, utilizing Python’s optimized factorial function.

## Method 2: Iterative Factorial Implementation

If for some educational or constraint reasons you can’t use the `math`

library, you can implement factorial calculation iteratively. This method manually computes the factorial by iteratively multiplying integers from 1 up to `n`

.

Here’s an example:

def number_of_ways(n): result = 1 for i in range(1, n + 1): result *= i return result # Example usage: arrangements = number_of_ways(4) print("Number of arrangements for 4 rooks:", arrangements)

Output: Number of arrangements for 4 rooks: 24

This code defines a function `number_of_ways`

that calculates the total number of non-attacking arrangements of rooks by multiplying integers from 1 to `n`

. It’s simple and requires no external libraries, making it a portable solution.

## Method 3: Recursive Factorial Implementation

Another way to compute the factorial is through recursion. This method calls itself with decreasing values of `n`

until it reaches the base case of `n`

equals 1.

Here’s an example:

def number_of_ways(n): if n == 1: return 1 else: return n * number_of_ways(n - 1) # Example usage: arrangements = number_of_ways(4) print("Number of arrangements for 4 rooks:", arrangements)

Output: Number of arrangements for 4 rooks: 24

In this recursive function `number_of_ways`

, the base case returns 1. If `n`

is greater than 1, the function calls itself with `n - 1`

, multiplying the return values. It’s elegant, but not as memory efficient for large `n`

due to the call stack limit.

## Method 4: Dynamic Programming

Dynamic programming can optimize the recursive factorial calculation by storing the intermediate results, thus avoiding redundant computation. This approach uses a bottom-up technique.

Here’s an example:

def number_of_ways(n): dp = [1] * (n + 1) for i in range(2, n + 1): dp[i] = dp[i-1] * i return dp[n] # Example usage: arrangements = number_of_ways(4) print("Number of arrangements for 4 rooks:", arrangements)

Output: Number of arrangements for 4 rooks: 24

The function `number_of_ways`

uses a list `dp`

to save computed factorials, calculating each number only once. It is a more efficient algorithm for very large `n`

as compared to the recursive approach.

## Bonus One-Liner Method 5: Lambda with Reduce

For fans of functional programming, this one-liner uses the `functools.reduce`

function together with a lambda to compute the factorial. The `reduce`

function applies the lambda cumulatively to the items of the given range, effectively calculating the factorial.

Here’s an example:

from functools import reduce number_of_ways = lambda n: reduce(lambda x, y: x * y, range(1, n+1)) # Example usage: arrangements = number_of_ways(4) print("Number of arrangements for 4 rooks:", arrangements)

Output: Number of arrangements for 4 rooks: 24

In this approach, the lambda expression in conjunction with `reduce`

from the `functools`

module creates a compact factorial function. It’s a neat one-liner, but potentially less readable to those unfamiliar with functional programming tenets.

## Summary/Discussion

**Method 1:**Factorial Calculation. The most straightforward and performance-efficient method using Python’s standard library.**Method 2:**Iterative Factorial Implementation. Provides a clear algorithmic understanding without library dependencies, good for educational purposes.**Method 3:**Recursive Factorial Implementation. Elegant, but less practical for large`n`

due to potential stack overflow.**Method 4:**Dynamic Programming. Excellent option for large`n`

, optimizing on the recursive method by storing intermediate results.**Method 5:**Lambda with Reduce. Compact and functional, great for one-liners but may sacrifice some readability.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.