Difference Of Permutation vs Combination With Examples

Many people are confused between the two terms permutation vs combination, but the difference is permutation differs from combination. Though they mean very similar things, there are a few key differences between the two terms. Both permutation and combination are important in counting. An arrangement of items in a specific order is referred to as a permutation. Combinations are the results obtained by choosing at least two items from a set of items on which it is not specified what number of items must be selected.

The difference between permutations and combinations can be understood by knowing the different conditions under which they are used. Our goal here is to explain how permutation and combination are different. Let us first define each of these terms separately before we look at the main differences.

What is Permutation?

Permutations are mathematical operations that are used to generate all possible combinations of a set. In general, people confuse permutations and combinations, thinking the two terms mean the same thing. There is a difference between the two terms - permutations are the variations that can be made from a collection of objects.

 

Permutation can be explained through its basic formula:

Basic Formula To Calculate Permutation

The formula for a permutation is: 

P(n,r) = n! / (n-r)!

where

n = total items in the set; r = items taken for the permutation; “!” denotes factorial

What is Combination?

Combination refers to the practice of selecting objects from a bulk collection in such a way that the selection method does not matter (non-similar permutations). The number of possible combinations will be able to be counted in minor cases. As a general rule, combination refers to the combination of n things taken one at a time without repetition. Combinations consist of r things selected from a set of n, where there is no need to replace them, and where the order of the things is irrelevant.

In order to understand the Combination, let's look at its basic formula:

Formula To Calculate Combinations

A permutation is calculated as follows: 

nCr = n! / r! * (n – r)!

The total number of items is represented by n, and the number of items chosen at a time is shown by r.

Difference Between Permutation vs Combination

Permutation and combination have different properties, so you need to understand the difference between them. A permutation refers to the many ways in which different items can be arranged. Combinations are smaller groups or sets created from a larger group. The arrangement of the individual parts within the group is not considered, just the collection of items that make up the collection. See the table below for a better understanding of Permutation vs Combination.

Permutation	Combination
Permutations are utilized when the sequence of arrangement is required.	Combinations are utilized to find the number of potential collections which can be formed.
Permutation of two from three given things x, y, z is xy, yx, yz, zy, xz, zx	Combination of two things from three given things x, y, z is xy, yz, zx
Permutations are utilized for things of a different sort.	Combinations are used for items of a similar type.
For the possible arrangement of ‘r’ things taken from ‘n’ things is nPr=n!(n−r)!nPr=n!(n−r)!	For possible selection of ‘r’ things taken from ‘n’ things is nCr=n!r!(n−r)!nCr=n!r!(n−r)!

When n and r are given as inputs, the permutation value will always be higher than the combination value. Combinations include only the different subgroups, whereas permutations include all possible arrangements. Permutation always gives a higher answer than Combination.

Permutation vs Combination: Order does/doesn’t matter” and “Repeats are/are not allowed

There are four variations of "Order does not matter" and "Repeats are not allowed":

Permutations:

There are basically two types of permutation:

• Repetition is Allowed: It could be “444”.

Example: 4× 4× … (3 times) = 64 = 64 permutations

• It's not allowed to repeat a race: for example, the first three places in a running race. First and second place aren't allowed.

Example: what order could 15 pool balls be in?

After picking, say, the number “13” we can’t pick it again.

So, our first choice has 15 possibilities, and our next choice has 14 possibilities, then 13, 12, 11, … etc. And the total permutations are:

16 × 15 × 14 × 13 × … = 1307674368000

Combination: 

You can also combine two things (remember you don't have to do them in order):

• Repetition is Allowed: such as coins in your pocket (4,4,4,10,10)

• No Repetition: such as lottery numbers (3,6,9,12,27,30)

Conclusion

Our discussion above hopefully clarified the difference between combination and permutation. The two combinatorial methods are also explained with examples for your better understanding.

Numbers are included in both permutations and combinations. Permutations, however, are determined by their order. Combinations do not matter in which order the numbers appear. 

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