# What is the difference between permutations and combinations?

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What is the difference between permutations and combinations?

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For Permutations the order matters, for Combinations the order does not matter.

A permutation example is, a “combination lock” the combination ONLY works in a certain order 3-25-17; if you put it in backways 17-25-3, the lock stays locked.

A combination example is, you have 10 friends, pick 3 to go to a concert. The order doesn’t matter, the three you pick are going to the concert.

A permutation is number of different ways you can order a collection of objects. For example, the permutations of ABC are: ABC, ACB, BAC, BCA, CAB, CBA.

Combinations are all the different ways you can select some number of elements from a set, where order doesn’t matter. For example, all of the combinations of size two from the set ABC are: AB, AC, and BC.

Permutations are ordered, combinations are not.

Lets say you have the letters {A, B, C, D, E} and you want to make a subset of three of them.

In permutations, the set (A, B, C) is considered to be distinct from (B, C, A) and (C, A, B). Wheras in combinations, all three are considered to be the same.

So using this example of a three-element subset of a five-element set, there are 60 possible permutations, but only 10 possible combinations.

If you care about order it’s a permutation.
Combination: Did you eat dinner and lunch and breakfast today?
Permutation: Did you eat dinner then lunch then breakfast today?

POM: Permutaion = Order Matters

COD: Combination = Order Doesn’t

I don’t remember much from my high school math classes but I sure remember the mnemonics

The difference is that for permutations, order matters.

A combination is a set or perhaps a group. A grocery list is a great example of a combination. If you buy different foods, you have a different combination. But if you walk through the aisles in a different order and still collect all the same food, that’s the same. The order you pick up the food in doesn’t matter.

A permutation is a sequence. Fighting games use permutations of button inputs to represent different attacks. One attack might be left-down-up-x and another could be up-left-down-x. Even though the same combination of button inputs are used, it is a different permutation, so it does a different attack.