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Combinatorics, also known as Combinatorial Mathematics, refers to the subject of mathematics dealing with issues of selection, organization, and operation within a limited or discrete system.
- The closely related field of combinatorial geometry is also covered under combinatorics.
- It has various applications from logic to statistical physics, and from evolutionary biology to computer science.
- Combinatorics can be defined as the study of finite discrete structures.
- It is involved with the enumeration of element sets as well as the study of permutations and combinations.
- It defines mathematical relations and their features.
- The word "Combinatorics" is used by mathematicians to refer to a broader subset of Discrete Mathematics.
- It is commonly used in computer science to create formulae and for the estimation of algorithm analysis.
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Key Terms: Combinatorics, Combinatorial Mathematics, Permutations, Combinations, Probability, Enumeration, Computer Science, Algebra, Network Security, Linguistics, Pattern Recognition
Combinatorics Meaning
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Combinatorics is a mathematical branch which deals with the enumeration, permutation and combination of sets of elements and the mathematical relations which represent their properties.
- Combinatorics, or combinatorial mathematics, deals with issues of selection, organisation, and operation within a limited or discrete framework.
- Combinatorics is extremely important in computer science. Combinatorics methods can be used to predict how many operations a computer algorithm will require.
- In discrete probability study, combinatorics is also important.
- Combinatorics techniques may be used to enumerate probable outcomes in a uniform probability experiment.
- Combinatorics is simply about the number of ways of choosing certain objects out of a collection or their arrangement.
- For instance, combinatorics is used in many fields, including Communication networks, cryptography and network security.
Permutations and Combinations Detailed Video Explanation
Read More: Factorial Formula
Features of Combinatorics
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The following are some of the key characteristics of combinatorics:
- Counting the structures of the type and size specified.
- Determining when specific criteria can be met and assessing criteria parts, such as combinatorial designs.
- External combinatorics are used to find the "largest," "smallest," or "optimal" elements.
Combinatorics Example
Algebraic combinatorics is the study of combinatorial structures that arise from an algebraic idea or the application of algebraic techniques to combinatorial issues.
Solved ExampleQues. Determine the numbers with at least one of their digits as 7, between 99 and 1000? Ans. Numbers which are between 99 and 1000 are all known to be three-digit numbers. Thus, the Total number of 3 digit numbers with at least one of their digits as 7 = (Total numbers of three-digit numbers) – (Total number of 3 digit numbers in which 7 does not appear at all) = (9 × 10 × 10) – (8 × 9 × 9) = 900 – 648 = 252 |
Read More: Algebra
Applications of Combinatorics
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Some of the major applications of Combinatorics are:
Network Security, Cryptography, and Communication Networks
In communication networks and parallel and distributed systems, permutations are commonly employed. In these sectors, routing different permutations on a network for performance evaluation is a prevalent topic. Many communication networks need safe data transport, necessitating advancements in cryptography and network security. Because of the rising usage of internet information exchanges, this field has lately become extremely important.
Computer Design
Input-to-output permutations are taken into account while designing computer chips. Field programmable interconnection chips allow users to programme connections for whatever permutation they choose. The placement of logic gates is an important part of computer architecture design.
Computational Molecular Biology
Computational molecular biology is a branch of molecular biology that uses computers to solve problems. Atoms, molecules, DNAs, genes, and proteins are all examples of combinatorial and sequencing difficulties in this discipline. Under certain conditions, one-dimensional sequencing issues are basically permutation problems.
Languages
Because the components of both languages, such as sentences, paragraphs, programmes, and blocks, are arrangements of smaller elements, such as words, letters, and atoms, both natural and computer languages are strongly connected to combinatorics. A string-searching algorithm, for example, may rely on word and character combinatorics. Combinatorics is significantly used in the study of computability, or what we can calculate and how we do it.
Pattern Recognition
All of the aforementioned domains may be seen as special examples of pattern analysis in a general sense. Languages deal with patterns of phrases that can take many different forms, including visual pictures, auditory signals, and other physical variables found in engineering challenges, such as electrical, pressure, temperature, and so on. Combinatorics may be used to describe some of these patterns. Computer music is a particular application domain of acoustic signal combinatorics.
Operations research
Combinatorics is used in many optimization issues in operations research (OR). The work scheduling problem is fundamentally a sequencing problem that involves determining the sequence in which jobs should be done in order to save overall time, cost, and other factors. Jobs in a computer system, network, or processing facility are all possible here.
Simulation
Permutations and combinations can be used in a variety of simulations. In genetic simulations, permutations reflecting diverse genotype and phenotype relationships are used. Networks, cryptography, databases, and OR are some of the other domains that use permutations and combinations for simulations.
Scientific Breakthrough
In the process of generating possible solutions for some types of knowledge discovery issues, the production of combinatorial sequences may become required. We can construct combinatorial sequences of chemical/biological radicals, bases, and molecular compounds as candidate solutions at each stage of the process and choose the most likely ones based on specific rules and limitations. Graph theory and number theory, for example, may provide combinatorial sequences as possible solutions.
Solved ExamplesQues. Show the number of ways by which 5 students can be chosen from a group 12 students. Ans.The number of ways as required = 12C5 \(={^{12}}C_{7}=\frac{12!}{7!(12-7)!}=\frac{12! }{ 7!\times5!}\) = 792 Ques. Determine the number of straight lines that can be formed by 8 points, out of which 3 are collinear. Ans. The number of straight lines that can be formed is: 8C2 − 3C2 + 1 (general formula nC2 − rC2 + 1). |
Read More: Class 11 Mathematics Chapter 7 Permutations and Combinations
Permutations and Combinations
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The different ways in which objects from a set can be picked, often without replacement, to produce subsets are known as permutations and combinations. This selection of subsets is called a permutation when the pattern of selection is a factor; it is called a combination when the order is not a factor.
Permutations and Combinations can be defined as the ways which help to express a group of objects by arranging them in a certain order or by forming their subsets. It simply shows a variety of ways in which data can be arranged after creating their sets and subsets. Permutation and combinations are the very basic methods of counting in the branch of Mathematics.
- Permutations can be defined as an act of arranging numbers or objects in a type of order.
- Combinations are the ways of picking numbers or objects from a group such that the order of the objects does not matter.
- Permutations are simply arrangements, while combinations are selections.
- The concept of factorials are generally used to understand permutations and combinations.
What are Combinatorics Infograph?
| Related Topics to permutations and combinations | ||
|---|---|---|
| Difference between Permutation and Combination | Set Theory in Maths | Permutations and Combinations formulas |
| Combination Formula | Stirling formula | Involute |
Combinatorics Formula
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Combinatorics Formula can be explained as a union of both the Permutation and Combination concepts. Permutation and Combination have the following mathematical forms:
Formula for Permutation
The act of organizing all the elements of a set into some order or sequence, or rearranging the ordered set, is referred to as permutation. Permutation Formula is:
| \({^nP_r} = \frac{n!}{(n-r)!} \) |
Where,
- n = Total number of objects in a given set.
- r = Number of objects that is to be chosen from the given set.
Formula for Combination
Combination can be defined as choosing members of a set, despite the order. Combination Formula is:
| \({^nC_r} = \frac{{^nP_r}}{r!} = \frac{n!}{r!(n-r)!}\) |
Where,
- n = Total number of objects in a given set.
- r: Number of objects that is to be chosen from the given set.
Read Also: Class 11 Maths Permutations and Combinations Important Questions
Combinatorial Techniques
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There are a number of important combinatorial rules or principles that are well-recognised and applied. Each of these ideas has a special application. For enumerative purposes, the rule of sum, the rule of product, and the inclusion-exclusion principle are frequently utilised.
To show that two sets have the same number of elements, bijective proofs are used. Some of the Combinatorial Techniques and Rules are as follows:
Summation Rule
If there are a possible way to do something and b possible ways to do something else, and the two things can't both be done, then there are a + b total possible ways to do one of the things, according to the law of sum. In more formal terms, the size of the union of two disjoint sets equals the size of the sum of their sizes.
Product Rule
Another intuitive notion is the rule of product, which states that if there are a ways to do something and b ways to do something else, then there are a b ways to do both.
Principle of Inclusion and Exclusion
The inclusion-exclusion principle is a counting method for determining the number of elements in a collection of sets. This counting method prevents elements from being tallied more than once if they appear in multiple sets in the union. It takes into account the size of each set as well as the size of the intersections between the sets. When there are two sets, the number of elements in the union is equal to the sum of the number of elements in A and B, minus the number of elements in their intersection.
Principle of Pigeonholing
If items are individually placed in one of the b boxes, where a > b, then at least one of the boxes contains more than one item, according to the pigeonhole principle. This technique can be used to show the existence of a certain element in a set of elements having specific qualities. Consider a set of three gloves, for example. There must be two left gloves or two right gloves in such a set (or three of left or right). This is a pigeonhole principle application that provides information about the qualities of the gloves in the set.
Read More: Theoretical Probability
Things to Remember
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- Combinatorics is the process of counting objects in a dataset using enumeration, permutation, and combination strategies to produce correct findings, according to scientific standards and notions.
- A permutation is an arrangement of things in a collection in a specific order.
- In a collection, a combination is an unorganised grouping of objects. The objects can be put in any order.
- Combinatorics can be found in a wide range of domains, including Network safety, Communication networks and cryptography, Computational molecular biology, Linguistics, Recognizing patterns, Simulation, Databases and data mining, etc.
- Algebraic combinatorics is a field of mathematics where the interplay of combinatorial and algebraic methods is significant.
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| Other Important Links | ||
|---|---|---|
| Solving inequalities | Tautology | Mathematical Logic |
| Indeterminate Forms | Algebraic Operations On Complex Numbers | R-squared formula |
Previous Year Questions
- If nCr−1 = 28, nCr = 56, and nCr+1 = 70, then the value of r is equal to…? [KEAM]
- An eight-digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating…? [JEE 2014]
- If a, b and c are the greatest values of 19Cp, 20Cq, and 21Cr respectively, then…? [JEE 2020]
- Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and…? [JEE 2019]
- In a collage of 300 students, every student reads 5 newspapers and every newspaper…? [JEE 1998]
- Number of divisors of the form (4n+2), n≥0 of the integer 240 is…? [JEE 1998]
- The number of 4-digit numbers without repetition that can be formed using the digits…? [KCET 2020]
- If nC12 = nC8 then n is equal to…? [KCET 2017]
- The number of words that can be formed by using all the letters of the word PROBLEM only one is…? [KEAM]
- How many four-digit numbers abcd exist such that a is odd, b is divisible by 3, c is even…? [KEAM]
Sample Questions
Ques. What is Permutation? (1 Mark)
Ans. Permutations can be expressed as an act of arranging numbers or objects in a certain type of order.
Ques. What is the definition of combinatorics? (3 Marks)
Ans. Combinatorics is one of the several disciplines and fields of mathematics. It is concerned with numbers, primarily the counting of those numbers. It's used to count numbers and objects in a discrete and finite dataset based on the data's conditions.
- It is also known as the field of mathematics concerned with the enumeration, permutations, and combinations of items in order to answer a question or solve a problem.
- Outside of pure mathematics, combinatorics has a wide range of applications, including theoretical computer science, statistics, and numerous branches of science.
- Combinatorics often also refer to a larger subset of discrete mathematics which broadly incorporate graph theory.
Ques. What's the connection between Probability and Combinatorics? (2 Marks)
Ans. Probability is a method for determining the most likely path to achieve a specific objective. In other terms, it refers to the number of alternative outcomes. Combinatorics, on the other hand, is the branch of mathematics concerned with calculating the number of potential item combinations.
These items are part of a limited statistical data collection with specified limits. Combinatorics and Probability are both employed in the field of computer science since it is necessary to count specific items and determine the likelihood of them following a particular pattern.
Ques. Arrange the letters in the word TAMIL in the following order:
(a)The letter T is always found adjacent to the letter L.
(b)T and L are inseparable (2 Marks)
Ans. Let's treat LT as a single letter and maintain it that way. We now have four alphabets that may be put in a row in 4P4 = 24.
L and T can be swapped in their locations in two different ways. As a result, total configurations equals 4!2! = 48.
Ques. Determine the number of different ways a cricket eleven may be chosen from a group of 15 players if:
(a)You have no limitations in terms of what you can elect.
(b)Every time, a certain player is chosen.
(c)There is never a specific player picked. (3 Marks)
Ans. a) When there are no restrictions on what can be chosen. As a result, 15c11 provides you with an entire number of possibilities.
b) Because a certain player is always chosen. As a result of 15-1, we have 14c10.
c) We have 14c11 since no specific player is ever chosen.
Ques. Determine the number of 5-student committees that may be selected from a class of 25 students. (2 Marks)
Ans. Because we have to choose five out of twenty-five options. Therefore,
Ques. P, Q, R, and S, four pupils, sit at four corners of the classroom at random while playing a game. Determine the probability that Q is seated in the room's northeast corner. (2 Marks)
Ans. We know that a student can sit at four corners in 24 different ways. In addition, Q can sit in the northeast corner in six different ways (that's three!).
As a result, the probability that Q will sit in the room's northeast corner is equal to the number of ways Q may sit in the northeast corner divided by the total number of ways = 6/24 = 1/4.
Ques. An urn is made up of 30 red and 70 green balls. What is the chance of receiving exactly k red balls in a sample of size 20 if sampling is done without replacement (repetition is not permitted)? (3 Marks)
Ans. Assume A is the occurrence (set) of receiving precisely k red balls. To find P(A)=|A||S|, we need to find |A| and |S|. First, note that |S|=
. To calculate |A|, we must first determine how many different methods there are to pick k red balls and 20-k green balls. We've arrived at the following result by applying the multiplication principle. 
As a result,
Ques. Assume there are k persons in a room, and we know
(a)k=5 with a chance of 1/4;
(b)k=10 with a chance of 1/4;
(c)With the probability of 1/2, k=15.
(d)How likely is it that at least two of them were born in the same month? Assume that each month has an equal chance of occurring.
(e)What is the likelihood that k=10? Given that at least two persons have birthdays in the same month, what is the probability that k=10? (5 Marks)
Ans. (a) Let Ak be the event that at least two people out of k people have birthdays in the same month. We have
\(P(A_k)=1-{p^2\over 12^k} for k\varepsilon[2,3,4,.....,12]\)
Note that PAk = 1 for k>12. Let A be the event that at least two people in the room were born in the same month. Using the law of total probability, we have
(b) The second part of the problem asks for P(k=10|A). We can use Bayes' rule to write
Ques. The word 'OPTICAL' has to be re-arranged in a way that vowels always come together. In how many ways can this be done? (3 Marks)
Ans. The given word, ‘OPTICAL’ consists of 3 vowels: O, I, and A. These vowels can further be coupled together to form a single letter, like CLTP(OIA).
Considering it to be 5 different letters (with each letter being unique),
Thus, the number of ways = 5! = 5 × 4 × 3 × 2 × 1 = 120
The remaining 3 vowels (OIA) can further be arranged among themselves as well.
The number of ways required to arrange these vowels among themselves = 3! = 3 × 2 × 1 = 6
Therefore, the number of ways of arranging the word as per the question = 120 × 6 = 720.
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