Python itertools: Permutations vs Combinations

Python's itertools module provides specialized functions for combinatorial computing, most notably permutations() and combinations(). While both functions take an input collection and generate subsets of a specified length, the key distinction lies in whether the order of elements matters. permutations() treats different orderings of the same items as unique outcomes, whereas combinations() considers order irrelevant, returning only unique groupings regardless of sequence.

The Core Difference: Order Matters vs. Order Does Not Matter

The fundamental distinction between these two functions comes down to mathematics:

permutations() (Order matters): The arrangement ('A', 'B') is treated as distinct from ('B', 'A').
combinations() (Order does not matter): The arrangement ('A', 'B') and ('B', 'A') are considered identical, so only one instance is produced.

Neither function allows an element at a specific index to be repeated with itself unless the input iterable contains duplicate values.

How `itertools.permutations()` Works

itertools.permutations(iterable, r=None) returns successive \(r\)-length permutations of elements from the provided iterable.

The parameter r defines the length of each output tuple.
If r is not specified or is None, r defaults to the length of the iterable, generating full-length permutations.

import itertools

data = ['A', 'B', 'C']

# Generate 2-element permutations
result = list(itertools.permutations(data, 2))
print(result)

Output:

[('A', 'B'), ('A', 'C'), ('B', 'A'), ('B', 'C'), ('C', 'A'), ('C', 'B')]

Notice that both ('A', 'B') and ('B', 'A') are present in the output.

How `itertools.combinations()` Works

itertools.combinations(iterable, r) returns \(r\)-length subsequences of elements from the input iterable.

The parameter r is mandatory; omitting it raises a TypeError.
Elements are returned in lexicographic order based on the input order. If the input iterable is sorted, the combination tuples will also be produced in sorted order.

import itertools

data = ['A', 'B', 'C']

# Generate 2-element combinations
result = list(itertools.combinations(data, 2))
print(result)

Output:

[('A', 'B'), ('A', 'C'), ('B', 'C')]

Because order does not matter, ('B', 'A') is omitted because its items are already represented by ('A', 'B').

Direct Comparison

Feature	`itertools.permutations()`	`itertools.combinations()`
Order Significance	Order matters	Order does not matter
Duplicate Sets	Yields both `(x, y)` and `(y, x)`	Yields only `(x, y)`
Length Argument (`r`)	Optional (defaults to length of input)	Required
Result Count Formula	\(P(n, r) = \frac{n!}{(n-r)!}\)	\(C(n, r) = \frac{n!}{r!(n-r)!}\)
Size of Output	Always greater than or equal to combinations	Always smaller than or equal to permutations

When to Use Which

Use permutations() when sequencing is critical to the problem. Examples include calculating seating arrangements, route scheduling (like the Traveling Salesperson Problem), or solving password and anagram puzzles.
Use combinations() when grouping items where the sequence does not change the result. Examples include selecting a committee from a pool of candidates, dealing a hand of cards, or creating lottery number sets.