# What is Permutation & combination: Definition and 12 Discussions

In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set {1,2,3}, namely: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory.
Permutations are used in almost every branch of mathematics, and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences.
The number of permutations of n distinct objects is n factorial, usually written as n!, which means the product of all positive integers less than or equal to n.
Technically, a permutation of a set S is defined as a bijection from S to itself. That is, it is a function from S to S for which every element occurs exactly once as an image value. This is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f(s). For example, the permutation (3,1,2) mentioned above is described by the function

α

{\displaystyle \alpha }
defined as:

α
(
1
)
=
3
,

α
(
2
)
=
1
,

α
(
3
)
=
2

.The collection of all permutations of a set form a group called the symmetric group of the set. The group operation is the composition (performing two given rearrangements in succession), which results in another rearrangement. As properties of permutations do not depend on the nature of the set elements, it is often the permutations of the set

{
1
,
2
,

,
n
}

{\displaystyle \{1,2,\ldots ,n\}}
that are considered for studying permutations.
In elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set.

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1. ### Number of ways of arranging 7 characters in 7 spaces

The rightmost position has 3 possibilities: ##x,y,z## The remaining two letters are to be arranged in 6 spaces: ##\frac{6!}{4!}## Now the 3 can be placed in ##\frac{4!}{3!}## Total no of ways =$$3×\frac{6!}{3!}=12×30$$ $$OR$$ Since ##x,y,z## are three different boxes/variables, we can use the...
2. ### Prob/Stats Books on Combinatorics, Permutations and Probability

Hello! I am looking for textbooks to relearn Combinatorics, Permutations Combinations and Probability and also Matrix algebra( decomposition, etc). I had done these many years ago and the course/books provided to me at that time weren't that great. So I want to relearn this with a more...
3. ### License Plate Combinations: Clarifying the Math

I have a question and searched about at google and found an answer which I don't make sure. If there is 26 letters and 10 digits; my answer is: first letter: 1 way(which is A) second letter: 26 way third letter: 26 way first digit: 1 way(which is 1) second digit 1 way(which is 2) third digit: 10...
4. ### B Can You Calculate the Odds of These Unlikely Events?

Hi folks - I need some help with a tricky probability. Here's the situation: Let's say there are 4M internet users in Age Group A. (The total set) Of those 4M, there are 1,000 users who play a specific sport. Those 1,000 are spread evenly over 125 teams, so 8 players each. 1. What's the...
5. ### No. of positive integral solutions of fractional functions

I know how to find integral solutions of linear equations like x+y=C or x+y+z=C where C is a constant. But I don't have any idea how to solve these type of questions.I am only able to predict that both x and y will be greater than 243554.Please help.
6. ### Statistics Permutations and Combinations

Homework Statement The back row of a cinema has 12 seats, all of which are empty. A group of 8 people including Mary and Francis, sit in this row. Find the number of ways they can sit in these 12 seats if a) There are no restrictions b) Mary and France's do not sit in seats which are next to...
7. ### Permuations of identical items

Homework Statement There is a book with 2 volumes. Each volume exists in 3 different languages. Each language has 2 identical copies(total of 12 books). In how many ways we can arrange them on a shelf, with no restrictions and order of the volumes is irrelevant? Homework EquationsThe Attempt...
8. ### A permutation and combination problem

.The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 41) and (41, 0), is (1) 901 (2) 861 (3) 820 (4) 780 my attempt: for this to be true i know that sum of x and y coordinate should be 41 but i don't know how to proceed.
9. ### I How many combinations of unique arrangements are there?

If there was a 1 billion x 1 billion x 1 billion cube made of 3D pixel cubes, and half of them are black and half of them are clear/colorless, then how many combinations of unique pixel arrangements are there? Would the amount of shapes/objects in this cube be infinite? (Assuming the black...
10. ### How Can Students Be Arranged into Groups with Set Sizes?

Homework Statement There are 30 students in a class. In how many ways can we arrange them if : a)we must have three group, group one must have 5 students , group two 10 students and group three 15 students. answer=\frac{30!}{5!*10!*15!} b)we must have three group and all must have 10 students...
11. ### How Many 5-Digit Numbers from 4, 5, 6, 8, 9 are Divisible by 8?

Homework Statement From the numbers 4,5,6,8,9 we make 5 digits numbers (each number can be used only once). h)How many of these numbers are divisible by 8? The correct answer is 20 Homework Equations a number is divisible by 8 if the last 3 digits are divisible by 8 If the hundreds digit is...
12. ### Permutation & Combination of multiple duplicates

Let me phrase the problem in a general way. Given n objects in a set. All the objects can be categorized into k groups such that no two objects from different groups are identical. Objects in the same group are indistinguishable from each other within the group. Number of objects in each...