Permutations and Combinations
Permutations and Combinations is one of the chapters in the syllabus of IIT JEE, which holds a place of importance and is also simple to understand. The topic on Permutations and Combinations is also extremely useful as it is a pre-requisite for the chapter on Probability. They also lay the foundation for Binomial theorem. This chapter deals with the fundamental concepts of Permutations and Combinations as developed form elementary ideas. It is evident that in forming combinations we are concerned with the number of things each selection contains, whereas in forming permutations, in addition to selection, we also consider the order of the things which make up each arrangement. The chapter begins with the basic principles which talks about Rule of Sum and Rule of Product. The concept of Permutations and Combinations are discussed after the basic principles. The other diversified concepts of restricted selection and arrangement, division and distribution of objects and derangements are also discussed. Some useful tips given towards the end are really fruitful in problem solving.
The topics covered under this head are:
We shall discuss some of the heads in brief as they have been discussed in detail in the coming sections.
What exactly do you mean by Permutations?
In simple words, a permutation is a one – to – one correspondence of a set onto itself. A permutation is basically an arrangement of a set of objects in a particular order.
The total number of permutations possible on a set of n objects is given by n factorial written as n! Hence, if we have three balls then the total possible number of permutations is given by 3! = 6. The figure given below illustrates the six possible permutations of red, green and blue balls.
View the following video for more on permutations
Circular Permutations:Cyclic permutations or circular permutations are the permutations framed using one or more elements of a set in cyclic order. Suppose there are three students A, B and C. then according to the concept of cyclic permutation they can be arranged in the following manner:
This leads us to the formula of cyclic permutations which states that the number of ways of arranging n distinct items in a fixed manner around a circle is given by
ways of arranging n distinct items in a fixed manner around a circle is given by
What exactly do you mean by a combination?
Combinations and permutations though are related, but are quite opposite in nature. When the order of things holds importance then the concept of combination comes into picture. The possible number of ways of selecting r items out of a total of n items is given by nCr. The point to be noted here is that the items considered in this case are unordered. For example, if we wish to find the number of combinations of two elements out of a set of 4 elements then n = 4 and r = 2. So, total number of combinations is = 4C2 = 4!/ 2! 2! = 6.
Permutations vs. Combinations:
The concepts of permutation and combination are quite related to each other and due to this students often tend to make a mistake while attempting questions on these topics. The chief difference between the two concepts is that of order. When the order of items is important then permutations are used while if the order of items is not important then combination is used. A permutation in fact, may also be termed as an ordered combination.
What is the Derangement Theorem?
Derangement is also a kind of permutation in which the elements of the set are deranged i.e. the elements do not appear at their usual places. The total number of derangements of a set with n elements is given by
Dn = n! (1 - 1/1! + 1/2! - 1/3! +… + (-1)n 1/n!)
What is the multinomial theorem in permutation and combination?
The multinomial theorem is the generalization of the binomial theorem to more than two variables. It provides us the expansion of the expression of the form
+ …. + xk
, where n is assumed to be an integer
where n = n1 + n2 + ….. + nk
Let us discuss some of the illustration based on these topics:
Illustration: The number of arrangements of the letters of the word BANANA in which the two N’s do not appear together is given by ….?
Solution: The question requires that the two N’s should not appear together.
Hence, n(P) = 6!/3!.2! = 60
When the two N’s are together then n(P) = 5!/3! = 20
Hence, the total number of arrangements remaining = 60-20 = 40.
Illustration: A five digit number divisible by 3 is to be formed using the numerals 0, 1, 2, 3, 4 and 5 without repetition. In how many ways can this be done?
Solution: A five digit number divisible by three is to be formed using the given numerals. Since this is to be done without repetition, it is only possible if the sum of the digits is multiple of three.
The number of ways of using the digits 0, 1, 2, 4, 5 = 18.104.22.168.1 = 96
The number of ways of using the digits 1, 2, 3, 4, 5 = 22.214.171.124.1 = 120
Hence, the required number of ways is 120 + 96 = 216.
Permutations and Combinations are important from IIT JEE perspective as a number of Multiple Choice Questions are framed on this section. If we consider the chapter of Permutations and Combinations, Binomial Theorem and Probability together we can say that a good understanding of the concept can help a student to get 10-20 marks varying from one examination to the other. It is very important to master these concepts at an early stage as this forms the basis of your preparation for IIT JEE. askIITians forum is a platform where you can ask your queries on the topics like derangement formula, circular permutations with restrictions, beggars theorem, division and distribution of objects, division by distribution etc. Click here to get answers from our IIT experts.