Introduction

From my experience tutoring CompSci and Maths undergrads I have noticed that many initially struggle with visualizing permutations.

And sadly, many resort to memorizing approaches to solve permutation problems rather than actually grasp it.

I believe this is partly caused by the non-intuitive way in which permutations are taught to students.

My aim in this post is to demystify permutations and show a quick, easy and intuitive way to visualize them.

Note: '...' means "and so on"

Visualizing Permutations

Let's consider this typical permutation problem.

How many 4 letter words can I form with 26 letters of the Alphabet?

How to visualize this:

1.

Each word will have four letters. So create four spaces for each letter.

_ _ _ _

3.

Let's consider the first space. It will contain a single letter that can be any of the 26 letters of the alphabet. So there are 26 possibilities for the first space. Leave the other spaces blank for now.

A_ _ _ , B_ _ _, C _ _ _, D_ _ _ ,. . .,Z_ _ _

We have 26 possibilities so far.

Imagine that list of possibilities being places in a single block.

Block 1 = [A_ _ _ , B_ _ _, C _ _ _, D_ _ _ ,. . .,Z_ _ _]

Now, let's consider the second space.

Again, it will contain a single letter that can be any of the 26 letters of the alphabet. So there are 26 possibilities for the second space.

So for each of the 26 possibilities in Block 1, the second space can be filled by any of the 26 letters of the alphabet.

Imagine filling the second character of each 26 possible words in Block 1 with each letter of the alphabet.

Let's ignore the other spaces and leave them blank.

For example, if the second letter is 'A':

[AA_ _ , BA_ _, C A _ _, DA_ _ ,. . .,ZA _ _]

If it is 'B'.

[AB_ _ , BB_ _, CB _ _, DB_ _ ,. . .,ZB _ _]

And so on...

Since there are 26 letters of the alphabets, there will be 26 more possibilities.

I.e.

[AA_ _,...,ZA_ _] , [AB_ _,...,ZB_ _],[AC_ _,...,ZC_ _]

So how many possibilites do we have in total so far.

We started with a block of 26 possibilities, Block 1.

Block 1 = [A_ _ _ , B_ _ _, C _ _ _, D_ _ _ ,. . .,Z_ _ _]

We made 26 different "versions" of Block 1 with the second character for each new version of Block 1 starting with a new letter of the alphabet.

i.e.

Block 1 version 1 = [AA _ _ , BA_ _, C A _ _, DA _ _ ,. . .,ZA _ _]

Block 1 version 2 = [AB _ _ , BB_ _, C B _ _, DB _ _ ,. . .,ZB _ _]

...

Block 1 version 26 = [AZ _ _ , BZ_ _, CZ _ _, DZ_ _ ,. . .,ZZ_ _]

In other words, we have Block 1 "times" 26 versions.

But Block 1 contains 26 possibilities. So in total you'll have 26 possibilities for each 26 versions of Block 1.

Or in other words 26*26 = 26^2

4.

Let's call the previous block Block 2.

Extending this logic, filling in the third character and leaving the next one empty, you'll have 26 versions of Block 2.

Block 2 contained 26^2 possibilities, and we're saying that there are 26 versions of Block 2.

So this new block, Block 3, contains 26^2*26 = 26^3

And so on...

So there are 26^4 different ways you can make a four letter word with the 26 letters of the alphabet.

I've included a picture to help visualize this.

Note

The trick is to imagine filling in a space of a n-letter word, leaving the other spaces empty.

Grouping that sequence of words in a block, moving down "one line" in your mind, and filling each following letter in each block horizontally, and continuing to move one line down once you've filled in a block.

Then just count how many line you went down, and that'll be the exponent. The base will be the character space (in this case 26 letters of the alphabet).

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