Mathematical Utilities

def num_combinations(n, r):
    """
    Calculate number of combinations (n choose r).
    
    Parameters:
    - n: int, total number of items
    - r: int, number of items to choose
    
    Returns:
    - combinations: int, number of combinations
    """

def num_permutations(n, r):
    """
    Calculate number of permutations.
    
    Parameters:
    - n: int, total number of items
    - r: int, number of items to arrange
    
    Returns:
    - permutations: int, number of permutations
    """

def factorial(n):
    """
    Calculate factorial of integer.
    
    Parameters:
    - n: int, input number
    
    Returns:
    - factorial: int, n! factorial
    """

def vectorspace_orthonormalization(ary):
    """
    Orthonormalize vectors using Gram-Schmidt process.
    
    Parameters:
    - ary: array-like, matrix of vectors (columns are vectors)
    
    Returns:
    - orthonormal_vectors: array, orthonormalized vectors
    """

def vectorspace_dimensionality(ary):
    """
    Compute dimensionality of vector space.
    
    Parameters:
    - ary: array-like, matrix of vectors
    
    Returns:
    - dimensionality: int, vector space dimensionality
    """

from mlxtend.math import num_combinations, num_permutations, factorial
from mlxtend.math import vectorspace_orthonormalization, vectorspace_dimensionality
import numpy as np

# Combinatorics examples
print(f"C(10,3) = {num_combinations(10, 3)}")  # 120
print(f"P(10,3) = {num_permutations(10, 3)}")  # 720
print(f"5! = {factorial(5)}")  # 120

# Vector space operations
vectors = np.random.randn(4, 3)  # 4-dimensional vectors, 3 vectors
orthonormal = vectorspace_orthonormalization(vectors)
dim = vectorspace_dimensionality(vectors)
print(f"Original vectors shape: {vectors.shape}")
print(f"Vector space dimensionality: {dim}")