Together, Apart and in Given Places

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle before we start. Take the word daughters. Nine letters, all different, and three of them are vowels: A, U and E, okay? Now I ask you two questions. One: in how many arrangements are the vowels not all together? Two: in how many are no two vowels together? Same question in different words? Hmm, hold that thought. By the end of this video, you will see that the first answer is more than double the second. And in JEE, these together and apart questions come up often. So, let's begin.

0:41Together: the block method

So, first, objects that must stay together. We use the block method. Imagine you glue those objects together into one single block. From now on, that block behaves like one object. Okay? Step two: arrange this block along with all the other objects. The block counts as just one unit in the row. Right? But wait. Inside the block, are the glued objects stuck in one fixed order? No! They can swap places among themselves and still stay together. So step three: multiply by the arrangements inside the block. Students forget this last step all the time. Don't be one of them! Clear?

1:22Apart: the gap method

Now the opposite situation: some objects must never sit next to each other. Look at the figure. Six boys and four girls stand in a row, all of them different people, and no two girls may be together, okay? Step one: arrange the unrestricted objects first. Here, the boys. They are the blue boxes, B one to B six. The small numbers at the bottom right of each B are just labels. Six boys in a row: six factorial ways. That exclamation mark is the factorial sign. With me? Now see the small circles, each with a little hat mark inside. Those are the gaps where a girl can stand. Count them with me. Five circles between the boys, and one at each end: seven gaps. Right? So, six boys in a row make seven gaps. Both ends count, okay? Many students forget the two ends, and the figure even says it: the seven gaps, including both ends. Now tell me. Can two girls go into the same gap? No! Then those two girls would stand side by side. So, the last step: put the girls in different gaps, one girl per gap. Then a boy always stands between any two girls. The girls are different, so the order matters: seven P four ways. That is the P with a small seven at the top left and a four at the bottom right, in the figure. So the answer is six factorial times seven P four. The cross means times. That gives six hundred and four thousand, eight hundred. Right? Now, careful. Suppose the six boys sit at a round table instead of a row. Still seven gaps? No, no! Only six. At a round table there are no ends, so the two end gaps join into one. So, m objects in a row make m plus one gaps, but at a round table, only m gaps. Count the gaps carefully. That is the trap. Got it?

3:20Not all together, or no two together?

Now, back to the words in our puzzle. Not all together means the restricted objects do not form one single block. The easy way: count all the arrangements, then subtract those where they are all together. Total minus all together, simple. Okay? Now think. Take three vowels that are not all together. Can two of them still sit side by side? Yes, of course! Two can sit together, with the third one far away. That arrangement is not all together, but two vowels are touching. So no two together is a stricter condition, and for it we need the gap method. Right? So write this line down. No two together is not the same as not all together. The equals sign with a slash through it means not equal. Mixing these two up is a very common mistake, so read every question slowly. Clear?

4:11Objects in given places

One more method, for objects in given places. The rule: fill the restricted places first, then fill the rest freely. Okay? Here is a quick one from the course. Seven people stand in a row, and one person, A, must be at neither end. In how many places can person A stand? Five places, correct! Seven places minus the two ends leaves the five inner places. With me? Now the other six people fill the remaining six places in six factorial ways. So the total is five times six factorial: three thousand six hundred. Restricted place first, the rest after. That's it!

4:52Example 1: EQUATION with the three consonants together

Alright, let's try one. In how many arrangements of the word equation are the three consonants together? All eight letters are different, and they stand in a row. Pause the video and try it with the block method. Done? Okay, let's check. First, sort the letters. The vowels are E, U, A, I and O, five of them. The consonants are Q, T and N, three of them, okay? Now glue Q, T and N into one block. The block and the five vowels make six units. Six units in a row: six factorial, seven hundred and twenty ways. So the answer is seven hundred and twenty, right? Oh no, careful! Remember step three. Inside the block, QTN can also be TQN, or NTQ, and so on: three factorial, that is six orders. So six factorial times three factorial, seven hundred and twenty times six: four thousand three hundred and twenty. Very good!

6:00Example 2: DAUGHTERS, vowels not all together, all apart

Now, remember our puzzle from the start? Let's finish it. The word daughters: nine different letters in a row, with three vowels, A, U and E, okay? In how many arrangements are the vowels not all together? And in how many are no two vowels together? Pause the video and try both parts. Done? Okay, let's check. Not all together first: total minus all together. Total: nine different letters, so nine factorial. That is three hundred and sixty two thousand, eight hundred and eighty. Now, all three vowels together. Which method do we use here? The block method, yes! Glue A, U and E into one block. The block and the six consonants make seven units: seven factorial. Inside the block, three factorial. Five thousand and forty times six: thirty thousand, two hundred and forty. So, not all together is three hundred and sixty two thousand, eight hundred and eighty, minus thirty thousand, two hundred and forty. Three hundred and thirty two thousand, six hundred and forty. With me? Now the second part. Is no two vowels together the same number? No, no, careful! That count still includes arrangements where two vowels touch. For no two together, use the gap method. Arrange the six consonants first: six factorial. In a row, they make seven gaps, both ends included. The three vowels go into different gaps, in order: seven P three, that is two hundred and ten. So, six factorial times seven P three: seven hundred and twenty times two hundred and ten. One hundred and fifty one thousand, two hundred. Now look back at our puzzle: about three hundred and thirty two thousand against about one hundred and fifty one thousand. More than double! Puzzle solved.

8:00Quick revision: block, gaps, not all vs no two together

So, quick revision. Together: glue them into a block, then multiply by the orders inside. Apart: arrange the others first, then put the restricted ones in different gaps, m plus one gaps in a row. Not all together is total minus all together, and given places are filled first. Now go and try the four practice problems on this page. You will find them easy, I promise. All the best!