Difference Between Permutation and Combination
The difference between permutations and combinations can be understood by knowing the different situations where the permutations and combinations concepts are used. For grouping of things, or to get a count of the number of subgroups that can be obtained from the given set of things we use combinations. And to find the number of possible arrangements of dissimilar things, we use permutations
The formulas of permutations and combinations are helpful to find the difference between permutation and combination. Here in this lesson, we can learn the important factors which help us to easily identify the use of permutations and combinations.
Difference Between Permutation and Combination
Difference between the permutation and combination is needed, to understand the right usage of permutation and combination. Permutation refers to the different possibles arrangement of things and is used when the things are of a different kind. And Combination refers to the number of smaller groups or sets which can be formed from the elements of a larger set. Here in combinations, we are only interested in the set of things which make a particular group, and the arrangement of the individual elements within the group is not considered. Let us check the below table to more clearly understand the difference between permutation and combination
Permutation  Combination 
Permutations are used when order/sequence of arrangement is needed.  Combinations are used to find the number of possible groups which can be formed. 
Permutations are used for things of different kind.  Combinations are used for things of similar kind. 
Permutation of two things from three given things a, b, c is ab, ba, bc, cb, ac, ca  Combination of two things from three given things a, b, c is ab, bc, ca 
For the different possible arrangement of 'r' things taken from 'n' things is \(^nP_r=\frac{n!}{(nr)!}\)  For different possible selection of 'r' things taken from 'n' things is \(^nC_r =\frac{n!}{r!(nr)!}\) 
For the given values of n and r, the value of permutation is always greater than the value of the combination. In permutations, the different possible arrangements are counted, but in combinations, only the different subgroups are counted. Hence the answer for permutation is always greater than the answer for combination.
Formulas of Difference Between Permutation and Combination
Difference between permutations and combinations can be understood with the help of the following formulas. Let us check the following formulas.
Factorial
The factorial function for a number is written as a natural number n, followed by a ! symbol.(n!). The factorial of n can be taken as the product of consecutive numbers 1, 2, 3, ... up to n. The concept of factorial is very useful to work across the formulas of permutation and combination.
n! = n × (n  1) × ......3 × 2 × 1
Permutations
A permutation is a count of the different arrangements which can be made from the given set of things. Let us take three characters P, 4, $. The number of different passwords which can be formed using these three characters is P4$, P$4, $P4, $4P, 4P$, and 4$P. This is a simple example of permutations of six passwords, which can be made from the three characters. The permutations of r things taken from n things are equal to the factorial of n divided by the factorial of the difference of n and r.
\(^nP_r = \frac{n!}{(n  r)!}\)
Combinations
A combination is all about grouping. The number of different groups which can be formed from the available things can be calculated using combinations. Let us try to understand this with a simple example. A man has four kids, Sam, John, Santa, and Mary. If he can take only three children at a time to the park, the possible ways he can take the three children would be:
 Sam, John, Santa
 Sam, Santa, Mary
 Sam, John, Mary
 John, Santa, Mary
These are the 4 different combinations or groups of 3 children who can be taken to the park. Thus the combination of 'r' things from the available 'n' things would be factorial of n, divided by the product of the factorial of r and factorial of (n  r).
\(^nC_r = \frac{n!}{r!.(n  r)!}\)
Relation Between Permutation and Combination
The formula of permutation and combination can be combined to form a single formula. The formula for 'r' things taken from 'n' things is such that their permutation is equal to the product of 'r' factorial and combination.
\(^nP_r = \frac{n!}{(n  r)!}\)
\(^nP_r = \frac{r!.n!}{r!.(n  r)!}\)
\(^nP_r = r! \times ^nC_r\)
Examples of Difference Between Permutation and Combination
The permutations and combinations have numerous applications in day to day line. Further, these concepts of permutations and combinations are also useful in probability. Some of the examples of applications of permutations and combinations are as follows.
 Seating arrangements for an important occasion or for official seating arrangements, the different possible seating arrangements can be calculated using the formula of permutations.
 The different number of passwords that can be formed from the given numbers, symbols, and alphabets are calculated using the permutation formula.
 Forming Teams from a larger number of prospective players is made convenient by knowing the different number of teams that can be formed. Here combinations are useful to find the number of possible teams which can be formed.
 Also, the different committees are formed by picking a few people, and the number of possibilities of forming a committee is calculated using the combinations formula.
Related Topics
The following links would help in a better understanding of the difference between permutation and combination.
 Permutation and Combination
 Probability
 Permutations
 Combinations
 Permutations and Combinations Formulas
Important Points
The following points help summarize the important learnings of the difference between permutation and combination.
 Permutations are applicable to find the count of the different number of arrangements that can be formed with the given things.
 The combination is useful to find the count of the number of different subgroups which can be formed from the given larger set.
 For the same values of n and r, the number of permutations(arrangements) is always larger than the number of combinations(groups).
Solved Examples on Difference Between Permutations and Combinations

Example 1: Find the different threedigit codes which can be formed using the digits 1, 2, 4, 5, 8, 9, by using the concepts from the difference between permutations and combinations.
Solution:
The given digits are 1, 2, 4, 5, 8, 9.
We need to form a threedigit code from the given five digits. Using the concepts from the difference between permutations and combinations, here we need to find the arrangements and hence we use the formula of permutations.
\(^nP_r = \frac{n!}{(n  r)!}\)
\(^5P_3 = \frac{5!}{(5  3)!} = \frac{5!}{2!} = 5 × 4 × 3 = 60\)
Answer: Therefore we can form 60 threedigit codes from the given 5 digits.

Example 2: In how many ways can a coach form a team of 2 players from among the six players in the academy? Try to use the concepts from the difference between permutations and combinations to find the possible solution.
Solution:
Here the aim is to select 2 players from the available 6 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.
\(^nC_r = \frac{n!}{r!.(n  r)!}\)
\(^6C_2 = \frac{6!}{2!(6  2)!} = \frac{6!}{2!.4!} = 15\)
Answer: Hence the coach can form 15 different teams of 2 players from the 6 players.
FAQs on Difference Between Permutation and Combination
What Is the Difference Between Permutation and Combination?
The permutation is the number of different arrangement which can be made by picking r number of things from the available n things. The combination is the number of different groups of r objects each, which can be formed from the available n objects. Further the difference between permutations and combinations can be clearly understood by checking few of its examples. Permutations are used for creating passwords, for creating different words from the set of alphabets, and for different seating arrangements. And the combination is used for the selection of people, formation of teams or committees, and a group of objects.
What Is the Formula for Permutation and Combination?
The formula for permutations is \(^nP_r =\frac{n!}{(n  r)!} \) and the formula for combinations is \(^nC_r =\frac{n!}{r!.(n  r)!} \). Before working and applying these formulas we need to understand n!. It is called n factorial and is the product of the consecutive numbers from 1 to n. Also we need to know that ^{n}P_{0} = 1, ^{n}P_{1} = n, ^{n}P_{n} = n!, ^{n}C_{0} = 1, ^{n}C_{1}= n, ^{n}C_{n} = 1.
What Is the Relationship to Find the Difference Between Permutation and Combination?
The formula for permutations is \(^nP_r =\frac{n!}{(n  r)!} \) and the formula for combinations is \(^nC_r =\frac{n!}{r!.(n  r)!} \). Combining both the formulas we can write \(^nC_r =\frac{^nP_r}{r!} \), or we have \(^nP_r =r!× ^nC_r \)
How Do You Find Factorial of a Number?
The factorial of a number is obtained by taking the product of all the numbers from 1 to n in sequence. Here we have n! = 1 × 2 × 3 × 4 × 5 × .......n. As an example let us find the value of 5! = 1 × 2 × 3 × 4 × 5 = 120. The formula of n! is used in the formulas of permutations and combinations
What Are the Examples of Difference Between Permutation and Combination?
The examples of permutations are for different arrangements such as seating arrangements, formation of different passwords from the given set of digits and alphabets, arrangement of books on a shelf, flower arrangements. And the examples of combinations are the formation of teams from the set of eligible players, the formation of committees, picking a smaller group from the available large set of elements.
Which of the Two Permutation and Combination Is of Greater Value?
The formulas of permutations and combinations is ^{n}P_{r} = n!/(n  r)! and ^{n}C_{r} = n!/r!(n  r)!. For the given value of n and r the permutations are greater than the combinations since the number of arrangement are always more than the number of groups which can be formed. Mathematically observing n! is the same in both the formulas, but the denominator in combinations is larger. Hence combination is lesser than permutations.
What Are the Topics in Mathematics Where Permutation and Combination Are Used?
The concepts of permutations and combinations are prominently used in probability, sets and relations, functions. The different sequences or arrangements can be found with the help of permutations, and the different groups can be found with the help of combinations.
What is 0!?
The value of 0! = 1 is used very often in formulas of permutations and combinations. Let us understand this with an example \(^nP_n =\frac{n!}{(n  n)!} =\frac{n!}{0!} =\frac{n!}{1}=n!\) , and \(^nC_n =\frac{n!}{n!.(n  n)!} =\frac{n!}{n!.0!} =\frac{n!}{n!.1}=\frac{n!}{n!} = 1\).
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