There are many different areas of mathematics, but the difference between the two terms that often confuse people are permutation vs combination. These two terms have very similar meanings but have some key differences. Both Permutation and combination are essential parts of counting. The word “permutation” can be defined as an arrangement of things in a particular order. A combination is the number of outcomes that results from each choice of at least two items taken from a set on which it has not been specified how many items are to be chosen.
The permutation and combination difference can be understood by understanding the different conditions where the permutation and combination concepts are used. Here we are going to explain everything about the difference between Permutation vs combination. But before we start explaining the main difference let us explain both the terms individually first.
What is Permutation?
A permutation is a mathematical operation used to generate all of the possible combinations of a set. The term permutations and combinations always gets confused, and people tend to think they are synonymous terms. Actually, they are not the same – while the word “permutation” refers to the number of arrangements that can be made from a group of objects.
Let us explain Permutation through its basic formula:
Basic Formula To Calculate Permutation
The formula for a permutation is:
P(n,r) = n! / (n-r)!
n = total items in the set; r = items taken for the permutation; “!” denotes factorial
What is Combination?
The combination is the way of choosing objects from a bulk collection, such that (non-similar permutations) the method of selecting objects doesn’t matter. We can say in more minor cases, we will be able to count the number of combinations. Combination relates to the combination of n things taken r at a time without repetitions. A combination is the selection of r things from a set of n things without any replacement and where order doesn’t matter.
Let us explain the Combination through its basic formula:
Basic Formula To Calculate Combination
The formula for a permutation is:
nCr = n! / r! * (n – r)!
Where n represents the total number of items, and r represents the number of items being chosen at a time.
Difference Between Permutation vs Combination
The permutation and Combination difference is required to know the right usage of permutation and Combination. When the items are of a different kind, permutation refers to their many potential arrangements. And the number of smaller groups or sets created from the bigger collection is referred to as a Combination. We are just interested in the collection of items that make up a group in combinations, and the arrangement of the individual parts inside the group is not considered. To further grasp the difference between permutation vs Combination, look at the table below.
|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)!|
For the given values of n and r, the permutation value is always higher than the value of Combination. The different potential arrangements are included in permutations, but only the different subgroups are included in combinations. Hence the answer for permutation is always higher than the answer for Combination.
Permutation vs Combination: Order does/doesn’t matter” and “Repeats are/are not allowed
4 variations of “Order does/does not matter” and “Repeats are/are not allowed”:
There are basically two types of permutation:
- Repetition is Allowed: It could be “444”.
Example: 4× 4× … (3 times) = 64 = 64 permutations
- No Repetition: for instance, the first 3 people in a running race. You can’t be 1st and 2nd.
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
There are also 2 types of combinations (remember the order doesn’t matter now):
- 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)
Examples of Difference Between Permutation vs Combination
Find the different three-digit codes, which can be formed using the digits 1, 2, 4, 5, 8, by using the concepts of permutations.
The given digits are 1, 2, 4, 5, 8.
We are required to form a three-digit code from the given five digits. Using the concepts of permutations, we need to get the arrangements and use the permutations formula.
Answer: Therefore, we can form 60 three-digit codes from the given 5 digits.
In how many ways can a coach form a team of 2 tennis players from among the six tennis players in the academy? Use the concepts of combinations to determine the possible solution.
Here the aim is to select 2 tennis players out of 6 tennis players. This is a case of forming a group, and hence we use the formula of combinations to find the possible number of teams that can be formed.
Here we have n = 6 and r = 2.
Answer: Hence the coach can form 15 different teams of 2 players from the 6 players.
We hope whatever we have discussed above is enough for you to understand the difference between permutation vs combination. We have also given examples of both permutation and combination for your better understanding.
Both permutation and combination include a collection of numbers. However, with permutations, the order of the numbers matters. With combinations, the order of the numbers does not matter.
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Frequently Asked Questions
What Is the Relationship Between Permutation and Combination?
The permutations formula is nPr=n!(n−r)!nPr=n!(n−r)! and the combination formula is nCr=n!r!.(n−r)!nCr=n!r!.(n−r)!. After Combining both permutation and combination formulas we can write nCr=nPrr!nCr=nPrr!, or we have nPr=r!×nCr.
What is the difference between a permutation vs combination?
The permutation is the number of different arrangements which can be made by choosing r number of things from the available n things. The combination is the number of different groups of r objects formed from the given n objects.