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# Mathematical Utilities
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Mathematical functions and utilities commonly used in machine learning computations and statistical analysis.
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## Capabilities
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### Combinatorics
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Functions for calculating combinations and permutations.
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```python { .api }
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def num_combinations(n, r):
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    """
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    Calculate number of combinations (n choose r).
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    Parameters:
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    - n: int, total number of items
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    - r: int, number of items to choose
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    Returns:
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    - combinations: int, number of combinations
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    """
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def num_permutations(n, r):
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    """
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    Calculate number of permutations.
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    Parameters:
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    - n: int, total number of items
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    - r: int, number of items to arrange
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    Returns:
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    - permutations: int, number of permutations
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    """
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def factorial(n):
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    """
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    Calculate factorial of integer.
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    Parameters:
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    - n: int, input number
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    Returns:
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    - factorial: int, n! factorial
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    """
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```
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### Vector Space Operations
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Linear algebra operations for vector spaces.
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```python { .api }
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def vectorspace_orthonormalization(ary):
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    """
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    Orthonormalize vectors using Gram-Schmidt process.
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    Parameters:
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    - ary: array-like, matrix of vectors (columns are vectors)
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    Returns:
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    - orthonormal_vectors: array, orthonormalized vectors
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    """
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def vectorspace_dimensionality(ary):
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    """
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    Compute dimensionality of vector space.
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    Parameters:
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    - ary: array-like, matrix of vectors
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    Returns:
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    - dimensionality: int, vector space dimensionality
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    """
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```
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## Usage Examples
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```python
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from mlxtend.math import num_combinations, num_permutations, factorial
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from mlxtend.math import vectorspace_orthonormalization, vectorspace_dimensionality
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import numpy as np
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# Combinatorics examples
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print(f"C(10,3) = {num_combinations(10, 3)}")  # 120
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print(f"P(10,3) = {num_permutations(10, 3)}")  # 720
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print(f"5! = {factorial(5)}")  # 120
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# Vector space operations
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vectors = np.random.randn(4, 3)  # 4-dimensional vectors, 3 vectors
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orthonormal = vectorspace_orthonormalization(vectors)
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dim = vectorspace_dimensionality(vectors)
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print(f"Original vectors shape: {vectors.shape}")
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print(f"Vector space dimensionality: {dim}")
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```