# Permutation and Combination in Python

Python provides direct methods to find permutations and combinations of a sequence. These methods are present in itertools package.

## Permutation

First import itertools package to implement the permutations method in python. This method takes a list as an input and returns an object list of tuples that contain all permutations in a list form.

## Python3

 `# A Python program to print all ` `# permutations using library function ` `from` `itertools ``import` `permutations `     `# Get all permutations of [1, 2, 3] ` `perm ``=` `permutations([``1``, ``2``, ``3``]) `   `# Print the obtained permutations ` `for` `i ``in` `list``(perm): ` `    ``print` `(i) `

Output:

```(1, 2, 3)
(1, 3, 2)
(2, 1, 3)
(2, 3, 1)
(3, 1, 2)
(3, 2, 1)```

Time complexity: O(n!), where n is the length of the input list. This is because there are n! permutations of n elements, and the program generates and prints all of them.
Auxiliary space: O(n!), as the program needs to store all n! permutations in memory before printing them out. Specifically, the perm variable created by calling permutations([1, 2, 3]) stores all n! permutations in memory as a list.

It generates n! permutations if the length of the input sequence is n.
If want  to get permutations of length L then implement it in this way.

## Python3

 `# A Python program to print all ` `# permutations of given length ` `from` `itertools ``import` `permutations `   `# Get all permutations of length 2 ` `# and length 2 ` `perm ``=` `permutations([``1``, ``2``, ``3``], ``2``) `   `# Print the obtained permutations ` `for` `i ``in` `list``(perm): ` `    ``print` `(i) `

Output:

```(1, 2)
(1, 3)
(2, 1)
(2, 3)
(3, 1)
(3, 2)```

The time complexity of this program is O(n^r) where n is the length of the input array and r is the length of permutations to be generated.

The space complexity is also O(n^r) as all permutations are stored in memory before printing.

It generates nCr * r! permutations if the length of the input sequence is n and the input parameter is r.

## Combination

This method takes a list and an input r as an input and return an object list of tuples which contain all possible combination of length r in a list form.

## Python3

 `# A Python program to print all ` `# combinations of given length` `from` `itertools ``import` `combinations`   `# Get all combinations of [1, 2, 3]` `# and length 2` `comb ``=` `combinations([``1``, ``2``, ``3``], ``2``)`   `# Print the obtained combinations` `for` `i ``in` `list``(comb):` `    ``print` `(i)`

Output:

```(1, 2)
(1, 3)
(2, 3)```

1. Combinations are emitted in lexicographic sort order of input. So, if the input list is sorted, the combination tuples will be produced in sorted order.

## Python3

 `# A Python program to print all ` `# combinations of a given length ` `from` `itertools ``import` `combinations `   `# Get all combinations of [1, 2, 3] ` `# and length 2 ` `comb ``=` `combinations([``1``, ``2``, ``3``], ``2``) `   `# Print the obtained combinations ` `for` `i ``in` `list``(comb): ` `    ``print` `(i)`

Output:

```(1, 2)
(1, 3)
(2, 3)```

2. Elements are treated as unique based on their position, not on their value. So if the input elements are unique, there will be no repeat values in each combination.

## Python3

 `# A Python program to print all combinations ` `# of given length with unsorted input. ` `from` `itertools ``import` `combinations `   `# Get all combinations of [2, 1, 3] ` `# and length 2 ` `comb ``=` `combinations([``2``, ``1``, ``3``], ``2``) `   `# Print the obtained combinations ` `for` `i ``in` `list``(comb): ` `    ``print` `(i)`

Output:

```(2, 1)
(2, 3)
(1, 3)```

3. If we want to make a combination of the same element to the same element then we use combinations_with_replacement.

## Python3

 `# A Python program to print all combinations ` `# with an element-to-itself combination is ` `# also included ` `from` `itertools ``import` `combinations_with_replacement `   `# Get all combinations of [1, 2, 3] and length 2 ` `comb ``=` `combinations_with_replacement([``1``, ``2``, ``3``], ``2``) `   `# Print the obtained combinations ` `for` `i ``in` `list``(comb): ` `    ``print` `(i) `

Output:

```(1, 1)
(1, 2)
(1, 3)
(2, 2)
(2, 3)
(3, 3) ```

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