How to study permutations and combinations for JEE mains and advanced

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.

Understand counting

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

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Studying Mathematics – should you know the derivation of formula

While teaching for IIT entrance, I usually come across two types of students; One that wants to know the derivation behind every formula, and the other that just wants to use the formula to get through the exam and tests. Why to reinvent the wheel, they ask. So what is important, knowing the derivation as well or just knowing the formula with their uses in problem solving?

This is not to suggest that you should or should not learn the derivation but you should be judicious in spending time on derivation. Remember that mathematics requires lot and lot of practice and then little more practice in the limited time.

There are two types of formulas. The first type is that which has been derived by large body of complex work. We see them in simple form for our use. This gives us an impression that the formula must have been simple to derive. This may not be the case. For example, long ago, a great mathematician, Bertrand, decided to solve a question. This is famously known as Bertrand Ballot problem. There were two candidates in an election. Candidate Martin got m number of votes while candidate Nathan won n number of votes. Here m is greater than n. When the counting happens, what is the probability that Martin is always ahead of Nathan. This means at any given point, the vote count of Martin is always more than the vote count of Nathan. This was certainly not an easy task to arrive at the probability of the event. After lot of deliberations, calculations, and recursion, Bertrand arrived at the probability will be

This looks pretty simple but deriving it will take lot of time. Whether you should understand the derivation of this depends on your level of preparation. If you have completed your syllabus and have nothing better to do than go to movies, I would suggest you try to understand this derivation. However, if you have chapters left in mathematics, move on.

The second types of formulas are the ones that are derived to offer us a simple form so that we can remember it and use it for problem solving. They are not difficult to derive but the purpose of making them ready made is that they are often used in problem solving. Hence using it saves time. These are like plug and play in computer. Do we really need to know the derivation? The answer depends on whether learning the derivation adds to our understanding of the subject at hand. For example, derivation of equation of a pair of tangents drawn on a circle from a point outside the circle is a good way to understand how a line is defined. The formula is SS1 = T2. However, understanding the derivation in this case will help you in defining your approach to problem solving. For example, knowing the derivation of all the ready made equations in coordinate geometry will help you master the subject.

So use your discretion when deciding to invest time in learning the derivation. Time is precious. This is the only asset that you students have. A judicious use of time will take you ahead from your competitors.

 

~ Pankaj Priyadarshi

Infinity – the final frontier of mathematics

In mathematics, we often come across statements such as “x tends to infinity” or “x -> ∞”. Most of the students read it and proceed to solve questions as per their understanding and defined process, tips, and tricks. Have you ever thought about infinity as concept? A great mathematician Rene Descartes (of Descartes rule of roots of polynomials) said that infinity is beyond the understanding of humans because we are finite. Whatever it meant the idea was to show that the very concept of infinity touches upon the existence of God.

Infinity – venture into unknown

Mathematically, infinity refers to a very large number, larger than any number you can think of. As soon as you think of any number, it is always possible to add a number and make it larger. There is no end to it. Hence statements such as n = ∞, or x takes the value of infinity really have no meaning. The correct statement is “n tends to infinity” which means n is a large number larger than any other number.

Essentially infinity refers to idea of unlimited such as space, time etc. Remember the quote of Einstein, “Only two things are infinity; the universe and the human stupidity. And I am not sure of the former”. At least, we humans have something in infinite supply.

Infinity as a concept creates some interesting situation in mathematics.

2 concentric circles with different sizes have the same number of points on the circumference. How? Draw a line from the centre intersecting the two circles. For every point common to the line and the bigger circle, you have a corresponding point on the smaller circle. Hence shouldn’t the number of points be same. There is no way to refute this.

2 sets of unlimited objects have the same size. Let’s say a set A is set of natural numbers while set B is set of real numbers between 1 and 2. Since both the sets have infinite number of elements, they have the same size. Who can refute this too?

We know that 2 parallel lines never meet. Well, here comes infinity… the place where they meet, far away from the madding limits of mathematics. Similarly asymptotes meet the hyperbola at infinity. Looks like infinity must be a nice place for meeting people torn apart by the limits of human mind. Our Bollywood too has adopted the concept of humanity where the young couple suffocated by traditions and limiting values swear that they will meet in next birth. This is the translation of mathematical infinity into something simpler to understand.

There are many more baffling examples.

You can create infinity too at some situation

We have infinity in our grip is when we draw the slope of tangent to a curve. Just draw a vertical line touching a curve and you have created a tangent with slope of infinity.

In a quadratic equation, put the coefficient of x2 equal to 0 and you have just created an equation with one root as ∞. Off course the second root is –C/A. If you put both the coefficient of x2 and x equal to 0, you have created a quadratic equation whose both roots are ∞.

Infinity, by its very nature, is unknown. The only person who ever tried to unravel it was none other than the extraordinary mathematician Ramanujan on whose life Hollywood made Oscar-winning movie called “The man who knew infinity” based on a book by the same name by Robert Kanigel. We can only try to understand a miniscule part of the infinite.

 

~ Pankaj Priyadarshi

Which subject to attempt first in JEE Advanced/Mains

For an IIT aspirant, this is a difficult question to answer, even more difficult than the JEE Advanced one. I am not kidding… just ask any IIT aspirant. In this article, I will try to answer this question in little details. This is solely based on my own successful experience long back as well as experience of my students who cleared JEE advanced and now studying in various IITs.

While I have mentioned JEE Adavanced in this article, this can be very well applied to JEE mains test too.

Before we attempt to answer this question, first answer this. If you are given 1 hr for each subject, what would you choose? I am sure you would look at your strongest subject and choose it because you can score maximum in the given time thus increasing your chances of success.

However, the real test does not divide time between subjects. This hard task is given to you to divide your time as you wish. Indeed, man is condemned to be free. In the real test, you are given 3 hrs and about 75 questions (roughly 25 from each subject). In my view, if you are reasonably good or strong at Chemistry, start your answers with chemistry. There is good reason for this. In the beginning, you should not only solve questions right but also maintain the good attitude for the next 3 hrs. Chemistry will help you achieve this.

At the same time, restrict your time to chemistry. Make sure that you do not spend more than 40-45 minutes on it. It is extremely important to restrict the time because you will need the saved time in mathematics and physics. Any extra time spent on one subject will make you regret when you have to leave few easy questions because of lack of time later. But if you are not good at chemistry, avoid it in the beginning as starting with it will frustrate you in the very beginning spoiling your chances in mathematics and physics. Start with the strongest subject.

Once you are done with chemistry, you can choose the relatively stronger subject out of Mathematics and Physics. If you choose Mathematics, again restrict the time to 1.25 hrs. Do not cross this because you will have only an hour for Physics. Then go for Physics. If you choose physics, follow the same timeline.

The reason for spending maximum time on your strongest subject (out of mathematics and physics) is that you can deliver more marks per extra minute, i.e. your marginal productivity will be much higher (remember marginal word from your application of derivative problems…). For example, if I spend 15 minutes extra in my strong subject, I can possibly solve 3 questions right while spending the same amount of time in other subject may make me score just 2 questions right.

Finally, do remember that JEE Advanced has individual cut-off apart from overall cut-off. Hence you cannot ignore any part of the question paper. Moreover, this advice is purely from my experience, if you have been taking JEE Advanced mock tests and have other ways to approaching different subjects, please continue with the same if you manage to score well. At the end of the day, all we want is your success in JEE.

 

~ Pankaj Priyadarshi

How to solve word problems

Many of you have come across problem which look more like English comprehension problem than mathematics riddle. Looks can indeed be deceptive. They usually come in the chapters of “Linear programing“, Application of derivative “,”application of differential equations.” etc. In some cases the problems may come from general topic such as function and relations, vector and 3D.

Here we are not considering problems from permutation and combination and Probability, coordinate geometry etc. because there are no other ways to represent these problems but by words.  Let’s look at how to proceed with word problems and how to work through their solutions.

Look for the key words

Words such as “slope”, ”tangents”, ”rate” etc. give you a clue that they essentially mean dy/dx or dp/dt etc. essentially ratio of change of one quantity with respect to another. It can also mean the rate of change of a quantity w.r.t. time (in case of rate). Translate the problem into mathematical language. For example if the question says “slope of tangent at a point is twice the y coordinate of line point”. change it to dy/dx=2y.

Similarly translate every relevant sentence to mathematics and voila, you got the problem in the language, you are comfortable with i.e. mathematics.

Look for the objective of the problem

Read the problem carefully and find what you have to answer. If you have to calculate the no of pens a company has to produce give it a name, say x. Remember that whatever you have to calculate or find are variable /unknown. Your every effort should go towards getting as much information as possible about the unknown variable.

Look for any constraints

Constraints are conditions that put limitation on variables. More constraints are good things because they limit the scope of the variable. For example if a problem says X ÎN, you have to  only consider natural number and not worry about rational, irrational, null, negative integers etc. This small condition limited the scope of your variable to just the natural number.

To summarize, every problem will have the following

objective :what is to be found.

Coustraints: the boundary within which you have to work

Other information: Help provided to let you proceed.

Combine these there factors and frame the question in mathematics .if you can do this, 80% of your problem in solved .Now do some calculation and make it 100%.

 

~ Pankaj Priyadarshi

Best Mathematics books for IITJEE Advanced

There are plenty of books in the market for IITJEE mathematics. However, abundance of books and authors in the market has only created more confusion. Here is my attempt to list few books which I found very helpful in IITJEE preparation. Take a look.

These mathematics books are mentioned for JEE Advanced purpose. The logic is simple. If you study for JEE Advanced, your JEE mains preparation is taken care of automatically. There are few topics which are in JEE mains but not in JEE Advanced. Even if they are in JEE Advanced syllabus, questions are rarely asked.

This is a comprehensive list. You may want to buy few of them or buy all. However, avoid being stingy on books. Books generally give you more than you pay for price.

 

First, few words about mathematics in IITJEE Advanced.

Mathematics in JEE Advanced is truly challenging for anyone. It tests your skill like no subject does. It is not that Physics and Chemistry in JEE Advanced are easier. Far from it, they can be equally daunting. But Mathematics questions can be extremely interesting as well as challenging. Mathematics offers variety of ways to form problems and IITJEE questions maker love to test your skill at problem solving. So here are few books that will help you tackle the JEE Advanced problems in a more confident way.

Arihant Books

Almost all the mathematics books of Arihant are great in the collection of problems. The problems are challenging, exhaustive, and tough, especially the level 2 problems. There is nothing better than to test your JEE Advanced problem solving skills than Arihant Mathematics books.

Coordinate Geometry and Calculus (both differential and integral) are simply awesome in scope and scale of problems. Algebra too is good but few chapters such as probability and permutation/combination could have been more exhaustive.

Cengage books

Cengage books are good. Explanation of concepts as well as problems is clear. As far as theory part is concerned, Cengage Algebra is better. Similarly 3D geometry and Vectors of Cengage is great to read for variety of problems. Calculus and Coordinate are equally good but the difficulty level of Arihant is slightly higher.

TMH JEE Advanced Mathematics

Another challenging book for JEE Advanced. There are plenty of problems in each chapter. Level 2 problems are really good because they test all your skills. The only problem is that there is no explanation of lever 2 problems. So if you are stuck with a problem, you have to rush to your coach.

Algebra by Bernard and Child

This book contains chapters from various topics, not necessarily Algebra. The problems are great and will be as challenging. The only problem is that the explanation of theory leaves much to be desired. Moreover, there is no help or hint in the book if you get stuck with a problem. This book is very useful.

Coordinate by Loney

Good book and has equally challenging problems but if you are doing with Arihant or Cengage, which should be fine.

Trigonometry: Any book by Cengage or Arihant will do.

Algebra by Amit Kumar of TMH

This book is best in Algebra. The content, types of problems, level of problems are all at the highest level. This is a gem we discovered while searching for JEE Advanced books.

Calculus: Thomas and Finney is good.

Calculus by Steward: This is again a great book. Level or problems are challenging and sometimes even beyond the scope of IITJEE.

A word of caution when reading books by foreign authors: The books are great to build sound understanding or mathematics as subject as well as problem solving skills. However, they cover many topics which may not be useful for IITJEE entrance. Hence study only those topics which are relevant.

 

~ Pankaj Priyadarshi

Advice to JEE advanced 2017 aspirants

If you score a position in top 2 lakhs students who will compete for the most coveted engineering colleges, i.e. IITs across the country. You will do a good job in coming this far, studying for board exam, JEE mains, and now ready for the toughest of them all, JEE Advanced. You will possibly have one and a half month to revise the important concepts, practice little more of problem solving, and practice doing it faster. Being a B. Tech. from IIT Kharagpur, I humbly think, that I am qualified to suggest few of the important things you need to take care of while doing your last 50 days preparation for JEE Advanced.

The first and foremost thing about JEE Advanced is that it requires thinking differently from the norm. In any other competition, you are given pattern based questions and you know how to approach the problem. In case of JEE Advanced, usually there are two types of problems;

The first type is completely “new”. New in this case means it requires new ways of thinking. This also means that whatever you have done in your board and JEE mains will support you to some extent but the fundamental thinking is something that you have to do on your own. Every year, IITs set up these kinds of problems to check your thinking and problem solving skills.

The second types of problems involve application of multiple concepts in a single problem. These types of problems make things difficult because you have to relate concepts from various branches of mathematics. Usually we do not study combination of mathematical branches while preparing. For example, when you solve vector problems, you do it from the book of vectors. Similarly, when you solve calculus, you do it by a calculus book. There is hardly overlapping between these subjects. There are few questions covering multiple subjects in a single problem which you can encounter. However, they are few and far between.

We will discuss some of the ways to do it in other article. But practice is the key. Go through all the previous years’ problems and solve on your own. If you are not able to solve them on your own, look at the solution and clearly indicate the lessons learnt from that problem. These small lessons learnt from various problems will help you tackle the next big problem and finally conquer JEE Advanced.

Best wishes…

~ Pankaj Priyadarshi

How to make the best use of 10 days between Board and JEE Mains

The biggest concern of students after the board is over is how to make the best use of the remaining 7-10 days to JEE mains. This year too, there is about a week’s time that students will get after their board exam. So how can you make the best use of your 7 days. Here are some guidelines.

Revise your strong points: This sounds contrary but revising your strong points will only make you more confident about scoring better in JEE mains. Remember that your strong points are the ones that will make all the difference. Make a list of formulas, exceptions, tricky problems etc. and go through it once more. It shouldn’t take you more than 2-3 days.

Revise 11th materials: Since you have just taken board exam, your 12th portion is taken care of. The knowledge is still fresh in your mind. Hence 12th is not concern. Go through 11th chapters and revise quickly. Here I am assuming that you haven’t forgotten everything that you studied in class 11th. If you have taken notes in your class or coaching, go through the notes. If your coach is still accessible, approach him or her for quick help on topics which you may need little help.

Avoid any new topic: This is not the time to learn new things but to consolidate what you already know. Learning new things will only take your valuable time which should be used for revision. For example, if you have ignored complex number till now, it will not help to waste time on learning it now. Time is too less for such a complex topic.

Take plenty of sleep and keep dehydrated: It is summer and water is crucial. Keep dehydrated. Sleep for 8-10 hours for a day or so just after the board to rejuvenate and take on JEE Mains.

Make a note of formulas and read them before going to bed: This is the best way to remember formulas and notes. Whatever you take at the end of the day, remains as fresh when you wake up in the morning. Do not think unnecessary thoughts before sleeping. It will only delay your sleep and eat into your valuable time.

 

~ Pankaj Priyadarshi

Advice to JEE Main aspirants – 2017

Now that the dates for JEE mains are announced, you all must be wrecking your brain out the preparation. Additionally the board exam in starting in march. The burden on your little may be too much to bear. While both exams are important and pressure is immense, you can take care of few important things to manage the pressure and keep your calm.

 

JEE mains and Board overlap on many areas of syllabus

Thankfully, there is overlap between the JEE mains and Board as far as syllabus, difficulty level and depth of questions are concerned. For example, a significant no. of questions in main is equivalent to that of Board. The only difference is that JEE main is a competition while board is an exam. In a competition, you are measured against the performance of others. In Board, your performance is absolute.

A thorough preparation for Board is foundation to score better in mains

The good news is that if you have prepared thoroughly for Board, you stand good chance to do better in JEE main. Students who have studied fundamentals pretty well will find JEE main relatively easier as it requires students to have thorough knowledge of fundamentals. Questions in JEE main rarely deviate from usual path unlike JEE Advance.

To prepare well for JEE main test, look at the syllabus at a micro level and list out the concepts which you are not clear about. Now attack them one by one. Take help from your coach if needed.

Practice is the key

While JEE main and board overlap in many areas, the need for practice in case of JEE main is more. Remember that JEE main tests you on objective questions with 4 options given. You have to select the right one in the given timeframe. Time is in short supply. Here apart from knowledge you also need speed. Speed in solving problems comes from practice. Practice as much as you can. Practice different types of problems; practice time saving technique which comes out of fundamental understanding of subjects. I can tell you this. If you take 2 students one with more talent and less practice will lose out to the one with more practice.

 

~ Pankaj Priyadarshi

A letter to board students

Dear Students,

The board exam for 12th starts now. This is the biggest test so far in your life. The last big one came about 2 years back when you appeared for 10th.

But this is certainly not the last challenge for you. There are tests, exams, competition far more challenging and far more important. They will appear as times go by.

However, 12th board exam is said to be the turning point in a student’s life. Your performance in board exam is a symbol to your future performance in most of the cases. So it is important to give your best to the exam and leave it to the almighty to reward you for your effort. I would like to share some of the important points to the students.

  1. Practice – there is no substitute to it

Bruce Lee, the legendary Kung Fu fighter said out that he is not afraid of a guy who has done 1000 shots but afraid of the guy who have practiced a shot 1000 times. So dear student, do it all over again; practice the problems; do the revision; remember that practice will help you achieve a great score.

  1. Focus – makes the practice perfect

Practice will not make you score unless you do it with focus. Focus is a great quality. It separates important from unimportant, relevant from irrelevant, and good from also ran, and finally great from good. Study with complete focus on books and tests. Throw away any distraction such as cell phone, iPod/iPad or any other object which distracts your attention. Two hours of focused study will make you a much better student than six hours of unfocused study.

  1. Faith – brings the results

Now that you have done your practice with focus, then what? This is where the faith comes in. Believe that you are doing great with your study. Have faith that your effort will pay handsomely when the results come out.

Finally, avoid negative thinking. You do more damage by negative thoughts than the good you build by positive thought. A positive frame of mind is vital. Remember that the toughest of them all i.e. JEE(Advanced) is next in line.

 

~ Pankaj Priyadarshi