Permutation and Combination
Permutation and combination are an important topic in Maths, and here we will try to understand the different situations where to use permutation or combination. In permutation the details matter, as the order or sequence is important. Writing the names of three countries {USA, Brazil, Australia} and {Australia, USA, Brazil) is different and this sequence in which the names of the countries are written is important. In combinations, the name of three countries is just a single group, and the sequence or order does not matter.
Permutations are useful to know the possible number of seating arrangements and also if one needs to know the different passwords which can be formed with the given digits. Further, if we wish to know the number of teams or the number of committees which can be formed from the available number of people, then we can make use of the concept of combinations. Let us learn more about permutation and combination in the below content.
Factorial
The factorial function for a number is written as a natural number n, followed by a ! symbol. It is written as 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
Example:
 3! = 3 × 2 × 1
 4! = 4 × 3 × 2 × 1
 4! = 4 × 3!
Permutations
A permutation is a count of the different arrangement 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 equal to the factorial of n divided by the factorial of the difference of n and r. Let us check the below formula of permutations.
\(^nP_r = \frac{n!}{(n  r)!}\)
Let us understand this concept of permutation with the help of a simple example. From a group of 5 students, we need to give 3 prizes to the students. To find the number of ways of giving these three prizes we can follow a stepbystep format. The first prize can be given to any of the 5 students, the second prize can be given to any of the 4 students, and finally, the third prize can be given to one of the 3 students. Thus by this method, the three prizes can be given in 5 × 4 × 3 = 60 ways. This can also be done using the permutation formula \(^5P_3 = \frac{5!}{(5  3)!} =60 \).
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. This is a simple example of combinations. This 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). The combination can be easily understood from the below formula.
\(^nC_r = \frac{n!}{r!.(n  r)!}\)
Let us look at another interesting example of the combination. Sam usually takes one main course and a drink. Today he has the choice of burger, pizza, hot dog, watermelon juice, and orange juice. What are all the possible combinations that he can try?
There are 3 snack choices and 2 drink choices. We multiply to find the combinations. 3 × 2 = 6. Thus Sam can try 6 combinations.
Difference Between Permutation and Combination
The difference between the permutation and combination can be understood through the following points.
 Permutations are used when order/sequence of arrangement is needed. Combinations are used when only the number of possible groups are to be found, and the order/sequence of arrangements is not needed.
 Permutations are used for things of a different kind. Combinations are used for things of a similar kind.
 The permutation of two things from three given things a, b, c is ab, ba, bc, cb, ac, ca. The combination of two things from three given things a, b, c is ab, bc, ca
 For different possible arrangement of things nPr=n!/(nr)!. For different possible selection of things nCr =n!/r!(nr)!
 For a given set of n and r values, the permutation answer is larger than the combination answer.
Related Topics
Solved Examples on Permutation and Combination

Example 1: Patricia has to choose 5 marbles from 12 marbles. In how many ways can she choose them?
Solution;
Patricia has to choose 5 out of 12 marbles.
Order doesn't matter here.
Thus she can choose it in \(12C_5\) ways.
\[\begin{align}12C_5&= \dfrac{12!}{5 ! \times(125) !}\\ &= \dfrac{12!}{5 ! \times 7!}\\&= \dfrac{12\times 11\times 10\times 9\times 8\times 7 !}{5 ! \times 7 !}\\&= \dfrac{12\times 11\times 10\times 9\times 8}{5 ! }\\&= 792\end{align}\]
Answer: Therefore there are 792 ways

Example 2: How many diagonals can be formed in the polygon?
Solution:
To form a diagonal, we need 2 nonadjacent vertices.
(\(\because\) 2 adjacent vertices will form a side of the polygon and not a diagonal).
The total number of ways of selecting 2 vertices out if n is \(nC_2\).
This number also contains the selections where the 2 vertices are adjacent.Thus, the total number of diagonals is \(nC_2  n\)
\(\begin{align}nC_2  n\\ &= \dfrac{n(n1)}{2}  n\\ &= \dfrac{n(n3)}{2}\end{align}\)
Answer: n(n  3)/2
FAQs on 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 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, a grouping 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 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 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 of 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 Areas 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 \(^nC_n =\frac{n!}{n!.(n  n)!} =\frac{n!}{n!.0!} =\frac{n!}{n!.1}=\frac{n!}{n!} = 1\).