# Permutation and Combination at Five Levels

**Level 1: Explanation for a Child**

Let’s think about ice cream. If you have 3 flavors - chocolate, vanilla, and strawberry - and you want to choose two flavors for your sundae, in how many ways can you do it? If the order matters, like you want chocolate on top of strawberry, that’s a permutation. But if you don’t care which one is on top, and you just want both flavors, that’s a combination.

**Level 2: Explanation for a Teenager**

Imagine you’re making a playlist of your 5 favorite songs. If the order of songs matters, it’s a permutation. For example, having “Song A, Song B, Song C” is different from “Song B, Song A, Song C”. However, if you just care about what songs are in the playlist and not their order, it’s a combination. So “Song A, Song B, Song C” is the same as “Song B, Song A, Song C”.

**Level 3: Explanation for an Undergraduate**

Permutations and combinations are two concepts in mathematics, particularly in statistics and probability theory. Permutations are about arranging items where the order is important. For instance, when you’re analyzing possible outcomes of a password composed of different characters. Combinations, on the other hand, are about selecting items where the order is not important. For example, in a lottery where you have to select a set of numbers regardless of the order.

**Level 4: Explanation for a Graduate Student**

Permutations and combinations are foundational concepts in combinatorics, used to calculate the number of possible arrangements (permutations) or selections (combinations) of a set of items. These notions are vital in many fields including probability theory, statistics, and computer science - from analyzing the complexity of algorithms to understanding random processes and statistical distributions.

**Level 5: Explanation for a Colleague**

Permutations and combinations are critical for combinatorial analysis, often serving as base cases in more complex combinatorial constructs. They’re employed in various computer science applications like generating permutations for backtracking algorithms, or evaluating combination possibilities in machine learning feature selection. Understanding the difference between permutations (order matters) and combinations (order doesn’t matter) is essential when dealing with problems of arrangement, selection, and distribution in combinatorial optimization and probabilistic modelling.

## Richard Feynman Explanation

Let’s imagine that you have a small box of various nuts and bolts. Now, suppose I ask you to take out three items from the box. There are many different ways you could do this, right? You might take out two bolts and one nut, or one bolt and two nuts, or three bolts, or three nuts, and so on. Each of these different possibilities is what we call a “combination”.

Now, suppose I ask you to arrange these three items in a line. For instance, you might put a bolt first, then a nut, then another bolt. Or you might put a nut first, then a bolt, then another bolt. Each of these different ways of arranging the items is a “permutation”.

So, in simple terms, a combination is about choosing items without regard to order, while a permutation is about arranging items in a particular order.

In the same way that we use mathematics to calculate the number of possible outcomes when we roll dice or flip coins, we can use mathematics to calculate the number of possible combinations or permutations. This can be really useful in all sorts of problems, from figuring out the odds of winning a lottery, to planning routes for deliveries, to analysing DNA sequences, and many others.

Just like how in physics, we often need to calculate probabilities - for example, the probability of an atom decaying, or a photon hitting a detector - in mathematics and computer science, we often need to calculate combinations and permutations. And although the math can sometimes get a bit tricky, the core concepts are really just about counting and arranging things.