Overview

Transcript

0:00Introduction

Hello everyone! Okay, a small challenge before we start. Take the six letters of the word school, and write all their arrangements in dictionary order, just like words in a dictionary. At what position does the word school itself appear? Oh, and one more thing: you get only two minutes. Sounds impossible, right? It is not. By the end of this video, you will do it well within two minutes. And look, permutations and combinations comes up often in JEE Main, so this speed really pays. So, let's begin.

0:36How to use this chapter

So, first, why this chapter? Permutations and combinations often gives one or two questions in a single JEE Main shift, okay? Those are marks you can win in a couple of minutes, if you are quick and careful. Now look at the kinds of questions. Numbers formed from given digits. Words, and their ranks in dictionary order. Committees. And the divisors of a number. Right? Then distributions, like balls into boxes. Derangements, where no object stays in its own place. And counting functions, or integers, with conditions. You have met every one of these in this unit. Clear? Now, the real JEE Main questions, from twenty twenty one to twenty twenty six, are already in this course. They sit at the end of Chapters one to five. The little dashes on screen mean to, okay? So this chapter does something different. Every problem here is original, but written in exactly the same style. Fresh problems, so you cannot simply remember an answer. Got it?

1:40Three sets, one clock

Alright, the problems come in three sets. Set one: numbers, words and arrangements. Set two: selections, divisors, and geometry, like counting lines and triangles from points. And set three: distributions, functions, and inclusion and exclusion, the dash here joining the two words. Three sets, and each one is a small mixed test. With me? Now the clock. The course says each problem should take one to two minutes. Not ten minutes, okay? One to two. Each set has fifteen solved problems. So tell me, how long should one whole set take you? Fifteen to thirty minutes, correct! Fifteen times one, and fifteen times two. So set a timer for thirty minutes, and sit as if it is the real paper. Right? And the most important rule of this chapter: attempt the whole set before reading any solution. Not one problem, then one solution. The whole set first, then check everything together. In the exam hall, no solution is waiting for you.

2:46In a mixed set, read the conditions first

Clear? Now, in a mixed set, nobody tells you which chapter a problem comes from. So what tells you which method to use? Yes, exactly! The conditions in the question. Every word matters. These phrases come straight from this chapter's problems. Repetition allowed, or all digits different? The count changes completely. Got it? Identical balls into different boxes, or different books into identical bags? Two completely different methods, okay? And people standing in a row, or sitting at a round table? Again, a different count. So, circle the conditions first, and only then calculate. Right?

3:28Example 1: lines through 15 points with 6 collinear

Now, one from set two, marked easy. Fifteen points lie in a plane, and exactly six of them are collinear, which means they lie on one straight line. And exactly means no other three points share a line, okay? How many lines do these points determine? Start your clock, pause the video, and try it. Done? Okay, let's check. Any two points fix a line. So we choose two points out of fifteen: fifteen C two, with the fifteen written up on the left of the C, and the two down on the right. That is one hundred and five pairs. With me? So, one hundred and five lines, right? No, no, wait! The six collinear points give six C two pairs, that is fifteen pairs. But all fifteen pairs lie on the same straight line. So that one line was counted fifteen times! So remove those fifteen, and add back the one line they really make. One hundred and five, minus fifteen, plus one: ninety one lines. And the trap? Forgetting the plus one, and writing ninety. Don't do that!

4:36Example 2: rank of SCHOOL in dictionary order

Okay? Now, remember our challenge from the start? Let's finish it. This one is from set one, marked medium. Find the rank of the word school among all the arrangements of its letters, in dictionary order. Two minutes on the clock. Pause the video, and try it. Done? Okay, let's check. Step one: write the letters in dictionary order. C, H, L, O, O, and S, okay? Notice, there are two O's. Now we count every word that comes before school. Words starting with C: the other five letters include the two O's, so five factorial over two factorial, the exclamation mark meaning factorial. That is sixty. And the same sixty for H, and the same for L as well. Three times sixty, the cross meaning times: one hundred and eighty. Right? Now the words starting with O, our repeated letter. Do we still divide by two factorial? No, we don't! And this is the trap. One O is used up in the first place, so only one O is left. The other five letters, C, H, L, O and S, are all different. So it is simply five factorial: one hundred and twenty. Divide by two factorial again, and the whole answer goes wrong. Clear? Now the words starting with S, the last letter. After S, school has C, the smallest letter left, and then H, again the smallest one left. So nothing new to count there, okay? The remaining L, O and O can be arranged in just three ways, and in dictionary order they are SCHLOO, then SCHOLO, and then school itself. Got it? So the rank is one hundred and eighty, plus one hundred and twenty, plus three. Three hundred and three! Challenge solved, and see, in well under two minutes. That is exactly the speed this chapter trains. Okay?

6:42Quick revision: timed sets and reading the conditions

So, quick revision. Three sets, with one to two minutes for each problem, so fifteen to thirty minutes for a whole set. Attempt the whole set first, then read the solutions. And read every condition first: repetition, identical or different, a row or a circle. Now open set one, start your timer, and begin. Then try the practice problems on each page. You are ready for this, I promise. All the best!