One of the most important topics for competitive tests such as JEE mains & advance, BITSAT and many others is permutations and combinations (P&C). In almost all the years, questions have been asked from this chapter. Needless to say, the chapter requires focused and detailed study to be able to solve questions in tests. Here are some of the important things you can do to be a master of permutations and combinations.
The most basic thing in mathematics is simple rule of counting. This simple concept suddenly gets an altogether different meaning when you study P&C. Most of the students simply skip this because it looks simple. Don’t get deceived by its simplicity. The basic rules of counting are pretty easy but practice a lot of questions to have a sound understanding of the concepts. This is the foundation of P&C.
Make the sample space small
Many times, the questions use big numbers. Counting using big numbers is of course difficult unless you use the formulae of P&C. Unfortunately, using the formulae doesn’t get you confidence. While you can use formulae to solve a question, do not forget to check your answer by using smaller numbers. It is always beneficial to use small numbers and counting to verify your answer.
For example, when you have to find out how many regions multiple intersecting circles divide a space into. Start with 2 circles, increase it to 3, then 4, you will see a pattern. It is easy to predict the answer when you increase the number by one and also derive the answer in terms of number of circles.
Learn and memorize the formulae
Fortunately, P&C has very few formulae, unlike the formidable coordinate geometry where they never seem to end. However, students do not memorize even these formulae thinking that they can derive in tests. This is flawed reasoning though. Derivation eats into your limited time which can be used to solve few more questions in JEE. Avoid this thinking and memorize what nCr / nCr+1 is equal to and any such variants. There are few more which must be remembered. They will help you not only in solving P&C problems but also binomial series.
Work on different variants of the problem
Divide the chapter into various types. For example, principle of inclusion-exclusion could be taken as one type; derangement is taken as another type etc. For each type, solve as many questions as possible, not the same type of questions but vary it as you wish.
For example, when you solve distribution problems, you generally come across two major cases. “Null value allowed” or “null not allowed”. Think of different cases. What if a variable can take value between 3 and 10 only? What does this extra condition change in the problem?
For example, when you solve questions on arrangement of elements of two types where no two should be adjacent to each other, think of the other cases. What if no three elements of the same type are adjacent to each other? How many ways are possible for this?
For example, when you find factors of a given number, think of a case when you want to find only even factors or odd factors. What changes you need to bring to your solution?
When you solve derangement problems, what happens when two of the elements are exactly identical? Does interchanging their places can be counted as one if we do not change others’ places?
Check your work once more
This is extremely important. Many times, students tend to count a specific case multiple times. This problem is also known as double counting. Check your answer for double counting as it inflates you answers and of course the answer will be wrong.
Finally, mind the difference between the various cases. Even a small change in the condition changes the complete question and ways of solving. Distribution problem and arrangement problems are notorious for putting new conditions to make problems tricky.
Work in these areas and see the change in your performance at IITJEE entrance.
~ Pankaj Priyadarshi