Permutations means different orders by which elements can be arranged. The elements might be of a string, or a list, or any other data type. It is the rearrangement of items in different ways. Python has different methods inside a package called itertools, which can help us achieve python permutations.
For example, if we have three balls – RED BLUE YELLOW
We can make different arrangements for this ball.
RED BLUE YELLOW
RED YELLOW BLUE
YELLOW RED BLUE
YELLOW BLUE RED
BLUE RED YELLOW
BLUE YELLOW RED
These all are possible arrangements where the order is essential and there is no repetition, and this is known as permutation.
Syntax of python permutations
Python has a package called ‘itertools’ from which we can use the permutations function and apply it on different data types. The number of total permutation possible is equal to the factorial of length (number of elements). In our case, as we have 3 balls, 3! = 3*2*1 = 6.
To import permutations() – from itertools import permutations
- Iterable – Here, we have to pass the iterable of whose permutations we want. Example of iterables- list, tuple, string, etc.
- Size- In this parameter, we have to specify the number of elements in each permutation. For example-
from itertools import permutations a=permutations ([1,2,3],2) for i in a: print(i) Output- (1, 2) (1, 3) (2, 1) (2, 3) (3, 1) (3, 2)
If we do not pass any argument in the second parameter, the default value is set as the length of the iterable.
from itertools import permutations a=permutations([1,2,3]) for i in a: print(i) Output- (1, 2, 3) (1, 3, 2) (2, 1, 3) (2, 3, 1) (3, 1, 2) (3, 2, 1)
You must be wondering why we are saving the result in a variable and printing the result using a ‘for’ loop. Let’s see what if we print the variable.
from itertools import permutations a=permutations([1,2,3]) print(a) Output- <itertools.permutations object at 0x00000265E51F1360>
We are getting this object as an output. So, we have to use a for loop to iterate through this variable and get the result.
Another way to get the output is making a list and then printing it.
from itertools import permutations print(list(permutations([1,2,3]))) Output- [(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)]
Using python permutations function on a string
If we want to find different orders in which a string can be arranged, we can use the permutations function. Let us see how-
string="ASHU" a=permutations(string) for i in list(a): # join all the letters of the list to make a string print("".join(i)) Output- ASHU ASUH AHSU AHUS AUSH AUHS SAHU SAUH SHAU SHUA SUAH SUHA HASU HAUS HSAU HSUA HUAS HUSA UASH UAHS USAH USHA UHAS UHSA
If we want to order these elements in the group of two, we can do the following-
string="ABC" a=permutations(string,2) for i in list(a): # join all the letters of the list to make a string print("".join(i)) Output- AB AC BA BC CA CB
You can notice that the total number of results are equal to the factorial of the size we are giving to 2nd parameter.
Find the order in lexicographical sorted order
If we want to find all the permutations of a string in a lexicographically sorted order means all the elements are arranged in alphabetical order and if the first element is equal then sorting them based on the next elements and so on.
from itertools import permutations string,n = input(“Enter string and size”).split() print(*[''.join(i) for i in permutations(sorted(string),int(n))],sep='\n') Output Enter the string and size BAC 2 AB AC BA BC CA CB
Using python permutations function on a list
Now, if we want to find all the possible orders in which a list can be arranged, we can use the similar approach as we did for string.
a=permutations([1,2,3,4],2) for i in a: print(i) Output- (1, 2) (1, 3) (1, 4) (2, 1) (2, 3) (2, 4) (3, 1) (3, 2) (3, 4) (4, 1) (4, 2) (4, 3)
We can also find the number of ways in which we can reorder the list using a single line of code-
print(len(list(permutations([1,2,3,4],4)))) Output- 24
Python Permutation without built-in function for string-
If we do not want to use the built-in function, we can make some function to achieve this goal.
answer= def permutation(string, i, length): if i == length: answer.append(''.join(string) ) else: for j in range(i, length): string[i], string[j] = string[j], string[i] #keep increasing i by 1 till it becomes equal to 0 permutation(string, i + 1, length) string[i], string[j] = string[j], string[i] permutation(list(string), 0, len(string)) string="ABC" print(str(answer)) Output- ['ABC', 'ACB', 'BAC', 'BCA', 'CBA', 'CAB']
Python Permutation without built-in function for lists-
def permutation(list1): # If the length of list=0 no permuataions possible if len(list1) == 0: return  # If the length of list=1, return that element if len(list1) == 1: return [list1] l =  for i in range(len(list1)): m = list1[i] # Extract list1[i] or m from the list. remlist1 is # remaining list remlist1 = list1[:i] + list1[i+1:] # Generating all permutations where m is first # element for p in permutation(remlist1): l.append([m] + p) return l if __name__=="__main__": print(list(permutation([1,2,3,4]))) Output- [[1, 2, 3, 4], [1, 2, 4, 3], [1, 3, 2, 4], [1, 3, 4, 2], [1, 4, 2, 3], [1, 4, 3, 2], [2, 1, 3, 4], [2, 1, 4, 3], [2, 3, 1, 4], [2, 3, 4, 1], [2, 4, 1, 3], [2, 4, 3, 1], [3, 1, 2, 4], [3, 1, 4, 2], [3, 2, 1, 4], [3, 2, 4, 1], [3, 4, 1, 2], [3, 4, 2, 1], [4, 1, 2, 3], [4, 1, 3, 2], [4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 1, 2], [4, 3, 2, 1]]
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There are some use cases or problem statements when we need to find all the possible orders in which elements can be arranged. So we use permutations from itertools. Some people get confused between combinations and python permutation, in permutations the order matters but in combinations, the order doesn’t matter.