**Introduction**

Mathematics is surrounded by many branches of vast areas like statistics, combinatorics, numerical analysis, probability, geometry, and many more. One of the known areas in mathematics is Combinatorics. Combinatorics deals with the arrangement and selection within a finite system or structure. It includes two most important parts- permutation and combination.

Permutation is how a group of objects can be arranged or selected. The order of the arrangements matters with the permutation. It can be understood by different examples and activities. Say a teacher provides a set of letters and asks the students to arrange them in various orders. The arrangement of the letters by the students in all possible ways is permutation.

While in combination, the order of selection does not matter. It is the way that determines the number of possible arrangements in a group of objects. Say, we have a set of 3 numbers P, Q, and R. The selection of the numbers from each set in as many as possible ways is a combination.

Permutation and combination are two different ways of grouping items of each set into sub-sets. In some scenarios, they share the same properties and ways. But based on terms, sequence, and concepts, they are different from each other. Let's understand the difference between them in detail.

**Permutation vs Combination**

The concepts of permutation and combination varying from each other are full of conundrums. The formulas and the meaning of both of these terms are a little confusing and seem like a riddle to solve.

Permutation simply means ordering and counting the arrangements of the objects. It focuses on the lists where order matters. The factorials of permutation involve all possible results of an event. The keywords of the concepts are- arrange, line up, and order. It uses the fundamental counting principle.

On the other hand, the combination means selecting objects from the group of the sets. It focuses on a selection of items in such a way that its ordering does not matter. In combination, the order of an arrangement is not important. It is used when the same type of items or objects are to be sorted. The keywords included in the combination are- choose, select, and pick.

Permutation and combination are the most fundamental concepts in Mathematics, which introduced the students to a new branch i.e., Combinatorics. So, it is necessary to differentiate them based on formula usage, value, key terms, and sample problems as well.

**Difference Between Permutation and Combination in Tabular Form**

Parameters of Comparison | Permutation | Combination |

Definition | Permutation refers to the number of ways to arrange a set of items in a sequence. | Combination refers to the number of ways to choose items from a large set. |

Derivation | Permutation is relevant and multiple permutations can be derived from a single combination. | Combination is irrelevant and a single combination is derived from a single permutation. |

Importance | The order in which the items are placed is important in permutation. | The order in which the items are selected is not important and does not matter in combination. |

Repetition | Permutation can be derived by repetition or without repetition of an object. | Combination does not involve repetition or non-repetition of an object. |

Formula | The formula of permutation is: _{n}P_{r = }n!/(n-r)! | The formula of the combination is: _{n}C_{r = }n!/r! (n-r)! |

Questions | Permutation raises the question of how many different arrangements can be created from a given set of items. | Combination raises the question of how many different groups can be chosen from a larger group of items. |

Examples | For example- in permutation, on tossing 3 coins HHT is a different permutation from HTH. | For example- in combination, HHT is the same as HTH. |

**What is Permutation?**

In simple words, Permutation is the arrangement of a set of objects in a particular way or order. It concerns the selection and arrangement of the objects. That is why permutation is also known as an ordered combination.

Permutation is denoted in many ways:-

- P (n,r)
- P
^{n}_{r} _{n}P_{r}^{n}P_{r}- P
_{n , r}

The elements of the sets are arranged in a sequence or linear order. A permutation is the choice of specific items from a set of items without replacement. In the permutation, the order of the objects matters and is important. It is used in every branch of mathematics and other fields of science. A permutation is used for analyzing and sorting algorithms in computer science. It is used to describe the states of the particles in quantum physics, and it is used to describe RNA sequences in biology. Permutation also comes in great use in daily life. There are many real-life examples of permutation, like arranging people, picking first, second, and third place, passwords, arranging digits, alphabets, colors, etc.

Permutation can be calculated using various formulas. The main formula is:-

P (n,r) = n ! ÷ (n-r)! , where n is the total items in the set, r is the items taken for permutation, and ! denotes the factorial.

The simple question that generalizes the expression of the formula is: How many ways can you arrange 'r' from a set of 'n' if the order matters?

Permutation can also be calculated without formula, by writing all possible permutations. The easiest approach is visualizing the permutation in many ways. A sequence of a 3-digit keypad can be arranged. Using the digits 0 through 9 and using a specific digit only once on the keypad, the number of permutations are:-

P (10,3) = 10 ! ÷ (10-3) ! = 10 ! ÷ 7 !

= 10×9×8

= 720

The formula of permutation can be derived in the following way:-

By the fundamental counting principle,

P (n,r) = n . (n-1) . (n-2) . (n-3) ---------- (n-(r-1)) ways

= n . (n-1) . (n-2) . (n-3) ----------- (n-r+1) --------- (1)

Multiplying and dividing (1) by (n-r) (n-r-1) (n-r-2) --------- 3. 2. 1

P (n,r) = [n . (n-1) . (n-2) . (n-3) ---------- (n-r+1)]

= [(n-r) (n-r-1) (n-r-2) -------- 3. 2. 1] [(n-r) (n-r-1) (n-r-2) -------- 3 . 2. 1 ]

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

Hence, derived.

There are five formulas for permutation in total, which are used in different situations:-

- Permutation formula without repetition- When ‘r’ thing from ‘n’ things have to be arranged without repetition.

^{n}P_{ r} = (n !) / (n-r) !

- Permutation formula with repetition- When 'r' from 'n' has to be arranged in the repetition.

n×n×n× _{--------- }×n (r times) = n^{r}

- Taken all at a time- When several ways of arranging 'n' among themselves are nothing.

^{n}P_{ n} = (n !) / (n-n) ! = n! o! = n! / 1 = n !

- Same sets of data- When all of 'n' things are not different and some of them are the same.

n! / (s_{1 }! × s_{2} ! × _{-------} × s_{n} ! )

- Circular permutation formula- When several ways of arranging 'n' different numbers of things in a circle.

(n-1) !

**Examples**

1. How many 4 letter words, with or without meaning, can be formed out of the letters of the word SWING when repetition of letters is not allowed?

**Answer: **n = 5

So, total permutations = P(n,r) = *5 ! 5-4!*

= *5×4×3×2×1×1*

= 120

Thus, 120 permutations are possible for the word SWING.

2. How many 2 letter words with or without meaning can be formed out of the letters of the word SMOKE when repetition of words is allowed?

**Answer: **n = 5

r = 2

P = n^{r}

= 5^{2}

= 25

Thus, 25 permutations are possible for the word SMOKE.

The fundamental counting principle states that if one operation can be performed in 'm' ways and there are 'n' ways of performing a second operation, then the number of ways of performing the two operations together is m×n.

This principle can be extended to the situation in which different operations are done in m, n, p -------- ways.

In this situation, the number of ways of performing all the operations one after the other is m×n×p×---------- and so on.

Some examples of the sample problems of permutations:-

1. Find the number of words, with or without meaning, that can be formed with the letters of the word PARK.

**Answer: **Total words= P (n,r) = 4!

= 4×3×2×1

= 24 words

Thus, 24 words can be formed with the letters of the word PARK.

2. A zip code contains 5 digits. How many different zip codes can be made with the digits 0-9 if no digit is used more than once and the first digit is not 0?

**Answer: **Since 0 is not allowed,

For the First possible total possible choices = 9

For the next 4 positions,

P = 9×P (9,4)

= 9× *9 ! 9-4!*

= 9× *9 !5 !*

= 9×*9×8×7×6×55 !*

= 27, 216

Thus, 27,216 zip codes can be made.

3. In how many ways can 5 different books be arranged on a shelf?

**Answer: **P = 5P5

= 5 ! / (5-5) !

= 5 ! / 0 !

= 5×4×3×2×1

= 120

Thus, in 120 ways, books can be arranged.

**What is Combination?**

Combination is a method of arrangement of the items where the order in which the items or the objects are chosen or selected does not matter. In other words, it means selecting things with no importance of their order. Apart from combinatorics, combinations are also studied in other fields like mathematics, finance, computer programming, probability theory, and genetics. In real life also, it is used in the selection of menus, making a cup of coffee, picking a table and selecting members, lottery games, selection of clothes, etc.

Combinations are also referred to as selections. The number of unique combinations out of a group of n objects is denoted by nCr. The general combination formula of the number of C of n different things taken r at a time is given:-

nCr = *n ! r ! n-r!* , where 0≤ r ≤ n

Another formula of combination is by using permutation:-

C (n,r) = P (n,r) / r ! where r is the size of each permutation, n is the size of the set, n,r are non-negative integers, and ! is the factorial expression.

The formula shows the number of ways a sample of 'r' items can be yielded from a larger set of 'n' distinguishable items.

The formula of combination can be derived in the following ways:-

C (n,r) = Total number of permutations/ number of ways to arrange r different objects

By the fundamental principle of counting,

Several ways to arrange r different objects in r ways = r!

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

= n ! (n-r) ! r!

Hence derived.

There are two types of combinations where the order of the objects does not matter:-

1. Combinations with Repetitions- Where repetition is allowed and the number of the items chosen is unlimited.

(n + r -1r)

2. Combinations without repetitions- Where order of selection does not matter, and each object can be selected only once.

rCn = *n ! r ! n-r!*

**Examples**

1. A father asks his son to choose 4 items from the table. If the table has 18 items to choose from, how many different answers could the son give?

**Answer: **r = 4

n = 18

C (n,r) = *n ! r ! n-r!*

= *18 !4 !18-4!*

= *18 !14 ! ×4!*

= 3060 possible answers

Thus, the son could give 3060 answers.

2. On the plane, there are 6 different points ( no 3 of them are lying on the same line). How many segments do you get by joining all the points?

**Answer: **C (6.2) = *62*

= *6!4 ! 2 !*

= *6.52*

= 15

Thus, by joining all the points, we will get 15 points.

3. If 18Cr = 18Cr +2 , find rC5.

**Answer: **18Cr = 18Cr +2

18C18-r = 18Cr +2

18-r = r + 2

2r = 18-2

r = 16/2

= 8

Then,

8C5 = *8 !5 !8-5!*

= *8 ×7×63×2×1*

= 56

**Main Difference Between Permutation and Combination (In Points )**

- Permutation is a way in which the ordering and counting of a set of items is done in a sequence. Combination is a way in which the selection of objects is done from the large sets.
- French mathematician and engineer Augustin–Louis Cauchy played a huge role in developing the theory of permutation. On the other hand, French mathematicians Blaise Pascal and Pierre de Femat contributed to the theory of combination.
- The word permutation is derived from the Anglo-French word 'permatacion', which means exchange or transformation. Whereas, the word combination is derived from the Late Latin word 'combinacyoun' which means the act of uniting in a whole.
- Permutations are used for things of different kinds. While combinations are used for things of similar kinds.
- Permutations are used for creating passwords, alphabets, and different seating arrangements. On the other hand, combinations are used for the selection of people, formation of teams, and groups of objects.
- The keywords like arrangement, order, and unique indicate permutation. Whereas, the keywords like selection, choose, and pick indicate combination.
- Permutation allows the repetition of objects. While combination allows the repetition of objects only in rare cases.
- In science and fictional novels, permutation can be symbolized as an example to depict the arranging of a gadget or others. On the other hand, the combination can be represented as the layers of various traits in an individual or others.

**Conclusion**

In short, permutation and combination are two different methods to take a set of items or options and create subsets. While permutation deals with arranging and ordering items, combination is the selection and choosing of items. Sometimes, these two terms become confusing. So, it is important to note that the problem of determining whether something is a permutation or combination depends on the given problems or conditions.