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What is Combinations and Permutations?

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Ahmed Shendy
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Today, I will discuss with you what is the concept of *Combinations and Permutations, and how to use them in programming.

I will be using Python to apply these concepts in programming, but language doesn't matter here, the concept is what matters, so whatever language you decide to use, there will be a library which gives you access to combinations and permutations.

In Python, itertools is the library that I use for combinations and permutations.

import itertools

Here is the main header in this article:

What is Combinations and Permutations

Both of Combinations and Permutatinos are mathematical concepts, and they are for finding all different/possible ways in which you can group list of items, but they differ in the order of items for each group:

  • In combinations the order does not matter.
  • In permutations the order does matter.

Apply Combinations and Permutations using Python

The methods combinations and permutations of itertools library receive two parameters, one is for interable object (e.g list), the second is how many items to include in each group.

Combinations Example:

We are looking for all different groups of 3 items from my_list.

my_list = [1, 2, 3]

combinations = itertools.combinations(my_list, 3)
for c in combinations:
    print(c)

(1, 2, 3)

Since the order doesn't matter, and my_list is a list of size 3, then we can only get one group.

So what about finding the different groups of 2 items?

my_list = [1, 2, 3]

combinations = itertools.combinations(my_list, 2)
for c in combinations:
    print(c)

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

Now we get more one group of 2 items, definitely they are more if the order matters, we will see it in permutations example.

Permutations Example:

Let's see all different groups of 2 items from my_list.

my_list = [1, 2, 3]

permutations = itertools.permutations(my_list, 2)
for p in permutations:
    print(p)

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

See? since order does matter in Permutations, (1, 2) will be different group than (2, 1), and we will get more groups than Combinations method.

Real Problems for Combinations and Permutations

So, let's see some real problems to learn how this might be important.

Problem:
Given list of positive numbers, print list of possible groups of 3 numbers so the sum of numbers in the same group equals 10.

Solution:
In this problem, order won't matter because both 1 + 3 + 6 and 3 + 1 + 6 will equal 10, so Combinations fits this problem.

my_list = [1, 2, 3, 4, 5, 6]

combinations = itertools.combinations(my_list, 3)

print( [result for result in combinations if sum(result) == 10] )

[(1, 3, 6), (1, 4, 5), (2, 3, 5)]

Problem:
Given a string of random letters, check if there is a group of these letters matches the word 'sample'.

Solution:
Order does matter here because as doesn't equal to sa, so Permutations fits this problem.

word = 'sample'
string = 'plmeas'

permutations = itertools.permutations(string, len(string))

for p in permutations:
  if ''.join(p) == word:
    print('Match!')
    break
else:
  print('No Match!')

Match!

That's all for today article, hope you like it.

Machine Learning