# Permutation vs. Combination

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## Difference between Permutation and Combination

“Permutation” and “combination” are two terms commonly used in everyday conversation, but they have special significance when used in mathematics. Two different concepts that are somehow related to each other, these two are the subjects of this comparison article.

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## Definition

While the word permutation as used in mathematics may refer to several different concepts, what they have in common is that they all involve “permuting”, which is the act of rearranging objects or values in an ordered manner. More precisely, permutation refers to the arrangement of a set of values into a certain order.

A combination on the other hand is a selection of several things from a larger group, although the order does not matter, unlike with permutations. The number of combinations could also be added in smaller cases. In a combination of fruits for instance, a combination can be made up of a pear and an orange, an apple and an orange, and a pear and an apple.

## In Use

Permutations are often quite prominent in various fields of mathematics. They often occur when different orders of finite sets are discussed, mainly because there is often a need to acknowledge such orderings and determine how many configurations there are. Permutations are also often considered when studying sorting algorithms in computer science. In fact, there is an entire field of algebra in which permutation are studied via the concept of symmetric groupings.

With combinations, it is also possible to find others that involve similar objects. As in the above example, there may be combinations made up of two apples, two oranges, or two pears. With larger sets, mathematics becomes necessary in order to determine the number of combinations. This is commonly seen in poker, where poker hands can be described as a combination of cards from the 52-card deck. In this case, the order of the cards does not matter.

## A "Key" Difference

As mentioned previously, the most important difference between permutations and combinations is the importance of order. With permutations, the order of events or objects is absolutely important, making one ordering distinct from another. In combinations, the order does not matter at all.

This difference can be more clearly explained with what is commonly known as a combination lock. This type of lock opens when you turn the numbered dials to specific numbers, forming a combination. However, the order of this number sequence is very important, otherwise the lock won't open at all. For purposes of this discussion then, a combination lock would be more accurately named a permutation lock. Of course, you probably wouldn't have much luck trying to market “combination” locks by that name!

## Similarities and Differences

Permutation

• Refers to the arrangement of a set of values into a certain order
• Often considered when studying sorting algorithms in computer science
• The order of events or objects is absolutely important

Combination

• A selection of several things from a larger group
• The order does not matter