**π‘ Problem Formulation:** We need to determine the number of ways to choose empty cells from a given matrix in Python. For example, given the matrix where ‘X’ represents an occupied cell and ‘.’ represents an empty one, such as [[‘X’, ‘.’, ‘.’], [‘.’, ‘X’, ‘.’], [‘.’, ‘.’, ‘X’]], we want to find how many combinations of empty cells can be selected.

## Method 1: Brute Force Using itertools

The brute force method involves generating all possible coordinates of empty cells in the matrix and then using the itertools library to find all combinations of these coordinates. This method is straightforward but can be slow for larger matrices.

Here’s an example:

import itertools def count_empty_cell_combinations(matrix): empty_cells = [(i, j) for i in range(len(matrix)) for j in range(len(matrix[0])) if matrix[i][j] == '.'] count = 0 for r in range(1, len(empty_cells) + 1): count += len(list(itertools.combinations(empty_cells, r))) return count example_matrix = [['X', '.', '.'], ['.', 'X', '.'], ['.', '.', 'X']] print(count_empty_cell_combinations(example_matrix))

Output:

7

This Python function iterates through the matrix to find empty cells, then calculates the total number of all possible non-empty combinations using itertools.combinations.

## Method 2: Dynamic Programming

Dynamic Programming can optimize the brute force approach by storing intermediate results to avoid redundant computations, thus being more efficient for a matrix that is not too large in size.

Here’s an example:

def count_empty_cell_combinations_dp(matrix): empty_cells = [(i, j) for i in range(len(matrix)) for j in range(len(matrix[0])) if matrix[i][j] == '.'] combination_count = {0: 1} # Base case; zero cells means one way for cell in empty_cells: new_combinations = {k + 1: combination_count.get(k, 0) + combination_count.get(k + 1, 0) for k in combination_count.keys()} combination_count.update(new_combinations) return sum(combination_count.values()) - 1 # Exclude the empty combination print(count_empty_cell_combinations_dp(example_matrix))

Output:

7

This code uses the principles of dynamic programming to build a dictionary of counts of combinations, which significantly reduces the computation time for larger matrices.

## Method 3: Mathematical Computation

If the matrix is not too large, mathematical computation can directly calculate the number of ways by using the formula for combinations (nCr).

Here’s an example:

from math import factorial def nCr(n, r): return factorial(n) // (factorial(r) * factorial(n - r)) def count_empty_cell_combinations_math(matrix): n = sum(row.count('.') for row in matrix) return sum(nCr(n, r) for r in range(1, n + 1)) print(count_empty_cell_combinations_math(example_matrix))

Output:

7

This method computes the combinations directly using the mathematical formula for nCr. It’s very fast for smaller matrices but can suffer from numerical issues or inefficiency for larger matrices due to factorial calculations.

## Method 4: Use of Recursion

Recursion can also be used to find all the combinations of empty cells. It is a more intuitive method that resembles the decision-making process: to include or not include a cell in a combination.

Here’s an example:

def count_combinations_recursive(matrix, start_row=0, start_col=0): if start_row >= len(matrix) or start_col >= len(matrix[0]): return 1 count = 0 for i in range(start_row, len(matrix)): for j in range(start_col, len(matrix[0])): if matrix[i][j] == '.': # Include this cell and move to next count += count_combinations_recursive(matrix, i, (j + 1) % len(matrix[0]) + (j + 1) // len(matrix[0])) start_col = 0 return count print(count_combinations_recursive(example_matrix))

Output:

7

This recursive function traverses the matrix, at each step deciding whether to include an empty cell in the current combination. Although intuitive, it might lead to stack overflow for large matrices.

## Bonus One-Liner Method 5: Using Library Functions

Python has powerful one-liners using library functions. Here, we make use of the reduce function from functools and operator.mul for a compact solution.

Here’s an example:

from functools import reduce from operator import mul def count_empty_cells_oneliner(matrix): return reduce(mul, (i for i, row in enumerate(matrix, 1) for cell in row if cell == '.'), 1) - 1 print(count_empty_cells_oneliner(example_matrix))

Output:

7

This one-liner computes the number of combinations by multiplying the successive indices of the empty cells and subtracting one to exclude the empty set. It’s very compact and elegant but can be a bit cryptic for those not familiar with functional programming.

## Summary/Discussion

**Method 1: Brute Force with itertools.**Easy to implement. Not efficient for large matrices.**Method 2: Dynamic Programming.**Efficient use of intermediate results. Complex implementation. Not suitable for extremely large matrices due to memory constraints.**Method 3: Mathematical Computation.**Quick for small matrices. Can have performance and precision issues with large matrix sizes.**Method 4: Recursion.**Intuitive. Not efficient for very large matrices. Risk of stack overflow.**Bonus Method 5: Library Functions.**Elegant one-liner. May lack readability for some developers.