Permutation in Python
A permutation refers to the number of possible arrangements of a set of objects where order matters. Permutations are widely used in mathematics, computer science, and programming problems that involve arranging data, generating sequences, or exploring all possible outcomes.
In this tutorial, we explain the mathematical concept of permutations and demonstrate how to generate permutations in Python using clear algorithms and practical code examples.
What Is a Permutation? | Mathematics Explanation for Python Kids
In mathematics, a permutation is an ordered arrangement of objects selected from a set. Because order is important, rearranging the same elements produces a different permutation.
For example, the arrangements ABC and BAC are considered different permutations of the same three elements.
Permutations commonly appear in:
- Combinatorics
- Algorithm design
- Password and key generation
- Game logic and simulations
- Search and optimization problems
In the unlikely scenario that the Teacher wants to see just how any four pupils, from a group of six (6), could be seated on a four-person desk; what this Teacher would be doing in essence is called Permutation (\(^nP_r\)).
Permutation Formula (\(^nP_r\)) | Maths Explanation for Python Kids
The number of permutations of selecting r objects from n distinct objects is calculated using the formula:
Where:
- n is the total number of objects
- r is the number of objects selected
- ! denotes factorial
This formula is useful when determining how many possible ordered arrangements exist before implementing a permutation algorithm in code.
Permutation With and Without Repetition | Maths Explanation for Python Kids
- Permutation without repetition: Each element can appear only once in an arrangement.
- Permutation with repetition: Elements may repeat, increasing the total number of possible arrangements.
The example below demonstrates permutations without repetition, which is the most common use case in programming exercises.
Permutation vs Combination in Python: What's the Difference?
Students often confuse permutations with combinations.
| Concept | Order Matters | Example |
|---|---|---|
| Permutation | Yes | ABC ≠ BAC |
| Combination | No | ABC = BAC |
It is easy to confuse permutations and combinations. The key takeaway is:
- Permutations (\(^nP_r\)): Use these when the order is important (e.g., a combination lock or race results).
- Combinations (\(^nC_r\)): Use these when only the group members matter (e.g., picking a committee).
When solving Python problems involving ordered arrangements, permutations must be used. If order does not matter, combinations are more appropriate.
Understanding this distinction is essential when implementing mathematical algorithms in code.
Generating Permutations Using Python
In programming, permutations are often generated by systematically rearranging elements in an array. Python provides a flexible environment for implementing permutation algorithms using recursion or backtracking.
The following approach demonstrates how to generate all permutations of an array in Python.
Python Permutation Algorithm (\(^nP_r\))
The Python algorithm for Permutation - \(^nP_r\), possible ways of arrangement - will simply be based on that of combination.
All that is needed after combination is a rotation or shuffle of
members of each possible combination result.
This shuffle simply involves interchanging the elements of the
combination group of size, r, to take all possible positions
starting from the extreme right to extreme left.
This is how our Permutation code in Python will work.
Create a new Python module file;
Call it Permutation.py
Type out the adjoining Python code for Permutation
(\(^nP_r\)).
Advice: You might want to keep the mother-class size (n)
and the group-size (r) small to avoid the Python permutation code taking too long.
As a rule-of-thump, DO NOT ASK QUESTIONS YOU DON'T WANT TO KNOW THE ANSWER TO.
Why Use Python for Combinatorics?
Using a Python math library or custom script allows you to build dynamic educational tools and interactive solvers. Our tool above uses this logic to give you instant results for any \(^nP_r\) calculation.
Applications of Permutations in Python Programming
Permutations are used in many real-world programming scenarios, including:
- Generating all possible test cases
- Exploring solution spaces in algorithms
- Cryptography and security
- Scheduling and optimization problems
- Educational simulations
Summary: Python Permutation Algorithm
Permutations are a powerful concept in both mathematics and programming. With these Python permutation tutorials, you can calculate \(^nP_r\), generate permutations of arrays or strings, and apply them to real-world problems.
For example, calculating possible arrangements in Python can help determine seating orders, password combinations, or sequence generation.
Python Code for Permutation - Module File
# define a class
class Transposition(Combinatorial):
def __init__(self):
super().__init__()
# till the ground for shuffle to grind on
def possibleWordPermutations(self, candidates, size):
self.perm_store = []
self.possibleWordCombinations(candidates, size)
# illegal 'r' value
if len(self.comb_store) == 0 or self.r == 1:
self.perm_store = self.comb_store
else:
last_two = [[], []]
for i in range(len(self.comb_store)):
self.index = self.r - 1
# copy up last two elements of 'comb_store(i)'
last_two[0] = [self.comb_store[i][self.index], self.comb_store[i][self.index - 1]]
last_two[1] = last_two[0][::-1]
self.index -= 2
self.local_store = []
self.local_store.append(tuple(last_two[0]))
self.local_store.append(tuple(last_two[1]))
if self.r > 2:
self.shuffleWord(self.local_store, i)
self.perm_store.extend(self.local_store)
return self.perm_store
def shuffleWord(self, arg_store, i):
self.local_store = []
for option in (arg_store):
members = []
members.extend(list(option))
# add 'index' 'comb_store[i]' element to this list of members
members.append(self.comb_store[i][self.index])
shift_index = len(members)
# shuffle this pack of words
while shift_index > 0:
# skip if already in store
if (tuple(members) in self.local_store) == False:
self.local_store.append(tuple(members))
shift_index -= 1
if shift_index > 0 and members[shift_index] != members[shift_index - 1]:
# interchange these two neighbours
members[shift_index - 1], members[shift_index] = members[shift_index], members[shift_index - 1]
# Are there any elements left? repeat if yes
if self.index > 0:
self.index -= 1
self.shuffleWord(self.local_store, i)
Python Code for Permutation - Main Class
from Permutation import Transposition
# Use the combination module/class
goods = ["Eno", "Chidi", "Olu", "Ahmed", "Osas", "Gbeda"]
perm = Transposition()
result = perm.possibleWordPermutations(goods, 3)
# print choices and operation
print("\n", perm.words, " permutation ", perm.r, ":\n")
# print out permutations nicely
i = 0
for group in result:
i += 1
print(i, ": ", group)
print("\n\nNumber of ways is ", len(result), ".\n")