🔥Last minute Revision of CSAT FORMULA in One video

Ajouté : 2026-05-23

[00:00:03]Hello and welcome you all. In today's session, we are going to revise all the CSET formulas. Ok?

[00:00:12]Now that is why I am recording this session today and then I will provide you the truth that now only three days are left. So which formulas should you revise in three days? What kind of formulas will come and what should you keep in mind before the CSET exam. Ok? So today I am going to discuss that entire formula with you.

[00:00:34]So let's begin. It is a very short session and if you watch it two-three times then there is no need to do anything more. Take a small A4 size paper.

[00:00:42]Keep a pen or pencil with you and keep writing. Let's start our session. First of all let's start with the number system. Because the number system is a very important topic for us. Now what things do you have to keep in mind in the number system? So the first thing that comes to mind is the concept of odd and even which I have taught many times.

[00:01:03]I have told you many times that what are the main things to keep in mind in this? So, all the points from one to eight have to be kept in mind very well. For example, if I add or subtract two even numbers, the result will be even.

[00:01:16]Same thing happens with odd, if there are two odd numbers, we subtract them and add them, then the result comes out to be even.

[00:01:24]But if one is even and the other is odd, what do I say, Sir, one is even and the other is odd, if you subtract or add, what will be your result? Odd will come. Ok?

[00:01:35]Next comes the fact that when you multiply two even numbers, the result will always be even.

[00:01:39]But even if one is even and the other is odd, it will still come out even.

[00:01:42]But if the odd is odd and you are multiplying, what will be your result? Odd will come. Ok?

[00:01:49]After that [sound of clearing throat] comes that if it is even then its power is not even or if it is odd then the result is always even. And odd is its power is even or odd. Does the result always come? Odd comes. So is this completely complete? How much did I say?

[00:02:02]Eight. Three related to subtraction and addition, three related to multiplication and two related to power. Ok? Is my point clear?

[00:02:08]You have to read this completely. This will raise one or two questions. Let's come next to the properties of rational and irrational. Well, Q stands for rational number and Q D stands for irrational number, you all must have seen the one shot session, so you must have come to know this thing.

[00:02:24]Now here there are two properties that if you add, subtract, multiply or divide two irrational numbers.

[00:02:33]What will always be the result? So the result that always comes can be rational or irrational. Ok? Here it is not defined whether it will be rational or irrational. Ok? So you have two irrational numbers. The result may be rational and irrational. But if one rationale becomes irrational and the other rationale becomes rational. q' stands for irrational number. So one became irrational and the other rational. If you are adding or subtracting, the result will always be irrational. But in case of multiplication and division, we will keep this thing in mind that the ration number which we are taking now should be non-zero. The ration here should be non-zero. Is my point clear? So what will be the ration number? will be non-zero. In case of multiplication and division. If this thing is followed then in all the four cases, the result is always irrational in the irrational and rational cases. Correct? So, these two things have been discussed. First is Odd Even and second is Ration Irrational property. Third, if you have to consider digits, if you have to take a two-digit number, then what will be its reverse like tan a + b? 10b + a Okay, let's take three digits a b c.

[00:03:46]What will be its reverse? So bac ok?

[00:03:50]This is the third thing. Then comes divisibility. You have to keep all this in mind brother. Keep this in mind.

[00:03:55]Next is divisibility. In case of two and five, check only the unit digit.

[00:04:00]In case of two, the unit digits should be 0 2 4 6 and 8. In case of five, it should be 0 and five. That's it. End.

[00:04:06]After that we come to the sum of digits in the case of three and nine, check whether it is divisible by three and nine or not. If this is happening then we can say that the number will be divisible by three and nine.

[00:04:16]In case of four, we will divide the last two digits.

[00:04:18]In case of 8, we will try dividing the last three digits.

[00:04:20]And now there is the matter of six.

[00:04:22]So any number that is divisible by two and three is also divisible by six. That's it. Now what did I say in this slide? So sir, let's talk about odd even.

[00:04:31]Eight things: addition, subtraction, multiplication and power. Correct? This [sound of clearing throat] has become odd and even.

[00:04:37]We have seen ration and personal property.

[00:04:39]After that, if the digits have to be considered, they can be two digit or three digit. I saw this also.

[00:04:43]In the case of two five, we have to check the unit digit, we have come to divisibility. So in case of 25, unit digit has to be checked. Well, in case of three and nine, check the sum of digits. As for the case of four and eight, if the last three digits of four are completely divisible by the last two digits of eight. what did you say? If it is completely divisible then we can say that in the case of four, the last two digits should be completely divisible.

[00:05:06]In case of 8, the last three digits should be completely divisible. So we can say that the number will be divisible by eight. Well, in the case of six, two and three both. Ok? Let's come to the next one, now let's talk about intake. So in the case of from, we have to do that the last unit digit is the right most digit. That unit digit has to be multiplied by two and subtracted from the rest.

[00:05:26]And how long will this process continue repeating again and again?

[00:05:31]Until a smaller number is found. From where we can tell by dividing whether the given number will be divisible by seven or not.

[00:05:37]If we talk about 11, then sum of digits at odd places minus sum of digits at even places. If the difference between these two either becomes zero or becomes a multiple of 11. Ok? What is written is that the difference between the two should either become zero or become a multiple of 11. So we can say that the given number will be divisible by 11.

[00:05:57]In the case of 13, what will you do with the unit digit? Multiply it by nine.

[00:06:01]What will you do with the unit digit? So sir, we will multiply it by nine.

[00:06:05]Ok? Multiply the unit digit by nine and subtract from the rest of the numbers. The unit digit is removed, multiplied by n and subtracted from the rest. Then the unit digit of the result was removed, multiplied by nine and subtracted from the rest. That's it.

[00:06:22]How long will this process last?

[00:06:23]Until a small number is found. Well, if a small number is divisible by 13 then we will say that it is completely divisible by 13. Another one told that in the common case of 7 11 13, take three digits from the last right most digit and subtract from the rest, till when? Until a smaller number is found. If the smaller number is a multiple of seven, then it will be said that it is divisible by seven. If it is a multiple of 11 then by 11. If it is a multiple of 13 then by 13.

[00:06:47]If the number is of six digits and all the six digits are repeating among themselves. Like I said 11 six times. Ok? So this is always divisible by 7 11 13. Well, numbers like 1 2 3 1 2 3 are also completely divisible by six 7 11 13.

[00:07:01]Next comes the composite number.

[00:07:05]What was said in composite number that it is a number which should have more than two factors. Well, if n is composite then it can be written as a product of primes.

[00:07:12]So the prime numbers we have p1 p2 p3 up to so on. Correct? Up to PN.

[00:07:16]Well, its power ranges from some alpha alpha alpha 3 to alpha n. Now what I am saying is that if you have to find the total number of factors.

[00:07:25]I have explained the meaning of factor multiple. Keep this in mind. So, the total number of factors is, what you do is just add one to its power and multiply them all and that will be your final answer. Well, if we want to find the total number of odd factors, then let us suppose that our prime number is what kind of prime number? So there is two, there is three, there is five, there is six, there is 11, there is 13 and up to so on.

[00:07:46]That's a lot of primes. So only one is even prime in it. So when will the factor become even? So just because of this prime, let's assume that if my P1 is two.

[00:07:56]P1 What is it? There is two. So just in case of odd number of factors, what you would do is remove this P1. Okay, right?

[00:08:06]Look here [sound of clearing throat] what have you written? If alpha1 = 1 else if p1 = 2 if p1 is 2 then just remove p1 and in rest this is the case of total number of factors which was adopted like alpha2 + 1 * alpha3 + 1 * alpha4 + 1 and up to so on. Similarly, if you want an even case then there are two things. First, if you subtract the total odd, you will get even.

[00:08:36]Secondly, write the power of two exactly. So this is the exact power of two. Meaning, what is written here? 2 to the power alpha1 * 3 to the power alpha2 If this composite number is broken down into these sub primes then prime factorization comes into play. Ok? So what is two to two? It is even. So there you will directly put the power of alpha and in the rest you will adopt the case which has the total number of factors.

[00:09:02]So what I said was that the exact power of 2 2 is the alpha 1 * alpha 2 + 1 * alpha 3 + 1 * alpha 4 + 1 and up to so on alpha n + 1. So what did I tell you here, divisibility by 7 11 13, this is the common divisibility of 7 11 13, then I told you about the composite number that the total number of divisors or factors, after that the odd factor, after that the even factor, how many total odd factors will there be, how many total even factors will there be, that's it, next comes the LCM. Let us find the LCM of three numbers.

[00:09:35]Meaning three types of numbers. The first one is integer. The second one is fraction and the third one is decimal.

[00:09:41]You have to keep all these things in mind.

[00:09:44]In the case of integers, primes were obtained by factorization.

[00:09:46]In case of fractions, LCM or HCF. In case of LCM, LCM of the upper one, HCF of the lower one.

[00:09:52]In case of HCF, the HCF of the top one is the LCM of the bottom one because the fraction will be in the form of a / b.

[00:09:56]Correct? Okay, convert the decimal into a fraction and then calculate further. So this LCM is complete. LCM is done in this slide. There is another method. What is another method?

[00:10:06]Meaning its application. What is the application? That is a number which I am dividing from a b c d to z. So corresponding to that some reminder like small a b c d is being generated. If the difference between the divisor and the reminder is constant. Let's say that's alpha.

[00:10:24]b - b is also alpha. C - C is also alpha. So in that case, the value of X that you will get is this value which you are dividing by capital ABCCFG. So its value is n times the LCM of this capital ABCD, meaning the LCM of the divisor is to be given as alpha.

[00:10:45]What to do now? Minus has to be converted to alpha.

[00:10:47]Your value will come. Well, why is n added to this? n belongs to a positive integer. Why is n used? That he will ask you to tell him the greatest three digit number which if I divide by 4 56, then my LCM, my reminder, is 1 2 3. Ok? So my reminder is 1 2 3 if I divide 4 by 56. Three digits have to be mentioned. So in this case the value of n can now be the LCM of two digits. It may be one digit.

[00:11:13]So to convert it into three digits, we have to multiply it by n. This means that by putting integers, one has to move towards bigger numbers.

[00:11:22]Secondly, Sir, the reminder was coming differently here. What if the reminder comes the same in every case, divided by a, still the reminder is coming. Divided by B but still the reminder is coming. Given by C, divide X, still reminder R and given by Z, still reminder R, R is coming in every case. So simply remove this alpha in this flower and replace it with plus R. What should be done with plus instead of this? R has to be done. Ok?

[00:11:46]Will you remove Alpha and what will you do in its place? plus R. That's it.

[00:11:50][sound of clearing throat] Over. So in this slide I have explained about LCM.

[00:11:53]Three types of integer, fraction and decimal. Convert the decimal to a fraction.

[00:11:57]In case of fractions, it is LCM of numerator, HCF of denominator.

[00:12:00]In case of HCF, HCF of numerator, LCM of denominator. What should he do? Have to convert. Here we have to do prime factorization. Ok? Next comes the case of HCF. In the case of HCF, the methods are somewhat different.

[00:12:13]Like before prime factorization you can adopt. Ok?

[00:12:17]What can we do after that? Long Divis Method. You can also try dividing it. I have told you the entire process. Watch One Shot once if you haven't seen it. Well, if any number of this type is coming. That a to the power m1 and a to the power n complete to -1 then if we want to find its HCF then that is equivalent to the a to the power HCF of m n - of 1 so if ab is a positive integer then HCF of ab * LCM of ab = a * b that's it.

[00:12:46]This is the case of HCF. Secondly, if we talk about cyclicity, it is very important. There is one number and there is a lot of power. How do you tell what its unit digit is? Or if you divide two by five by 10, what reminder will come? It is a very simple thing. There is a process of cyclicity. Just look. The cyclicity of 2 3 7 8 will be four divided by four to the power. The reminder that will be generated will be 1 2 3 and zero, put four in its place.

[00:13:11]Then apply all this power to the unit digit. And then the unit digit which will come in case of two, in case of five and in case of 10. If you want to give a reminder, just divide the unit digit, sorry, by the unit digit.

[00:13:25]What should the unit digit be divided by? Your reminder will come out from two to five and from 10.

[00:13:28]Well [sound of clearing throat] 0 1 and 6 and five it has a cyclicity of one.

[00:13:32]Take any power on this and the unit digit will remain the same. Four and no have a cyclicity of two.

[00:13:38]We can also understand this from odd-even that if the power is odd then what will happen? What will happen if the power is even? Try it once. Like what is 4 to the power 1 power? Odd. So what is the unit digit corresponding to that?

[00:13:48]What is 4 to the power 2? Even.

[00:13:50]What is its corresponding? The unit digit is coming.

[00:13:52]And after that it will start repeating.

[00:13:54]So we can understand it here through odd and even.

[00:13:56]Next come reminders. There is a basic formula for a good reminder.

[00:14:00]That is the dividend equals divisor * cosecant plus the reminder. Well, now you will understand that the multiplication has been given in such a way that a * b * c will divide it by x.

[00:14:10]Well, x is less than a < b < c, which means that the value smaller than the one above is a smaller value, so divide all of them, specifically divide a / x reminder ax b / x reminder bx c / x reminder cx Now all these values ​​are smaller than x, so multiply them all and divide by x, the reminder that will come from here, the same reminder will come from here, UPSC has asked this question three-four times, so keep this in mind, well, if the value of x becomes greater than a b c, then you will not be able to do this process, if the value of x is greater, then the same reminder that is a will come. Then what will you do? So whatever the common factor is, remove it. Okay, right? Meaning, x might be deducting something from the above, so that common factor was removed. Well, if the denominator has the value one, meaning the value of x after removing the common factor has become one, then what is the direct reminder? Zero. But if the denominator does not contain the value one. Apart from one. So, whatever factors you have removed from the top and bottom, you have to multiply them by the reminder that will come. So factor multiply by reminder initial means after cancelling out the factor, if you give division then some reminder will come.

[00:15:16]Whatever reminder I have generated, I will have to multiply it by the factor by which the factor has been removed from the top and bottom. Ok? That's it. Well, this is something related to polynomial theorem that 7 to the power 100 / 4, this type of problem has also been asked many times.

[00:15:29]Well, what will you do? We will break it into two parts, the truth is that the first part should be a multiple of the divisor, meaning it could be zero, it could be four, what else could it be? It could be eight, it could be 12, it could be 16, right? And what will we do with this, second part four to the smaller power 100? We will remove it and rest 2 re to the power 100 plus minus don't forget. Okay, right? And we will proceed further like this. If a small value comes, we will just send a reminder. That's it. Next comes.

[00:15:57]Next you have that number of zeros.

[00:15:58]In the case of number of zeros, what you have to do is just look at the powers of two and five.

[00:16:03][sound of clearing throat] Right?

[00:16:04]In case of number of zeros, what you have to do is look at the powers of two and five.

[00:16:08]Your number of zeros will be calculated.

[00:16:10]Meaning that the product of some consecutive numbers might have been given. There, pay two and five to the power of five. Just find out what is the maximum power of two?

[00:16:19]What is the maximum power of five? And whichever is the lowest among them. Let's get 2 to the power a.

[00:16:23]5 to the power b is here. If b is small then b is the number of zeros. If a is smaller then a number of zeros. Similarly, I had also told you in one short session that if 35 to the power X asks you if you have been given an executive product, what will you do? 35 will break it. 7 * 5 and find the power of a and b. Okay, right? Find the power of 5.

[00:16:42]How much is it? Find the power of to.

[00:16:44]Similarly, if there were 45, there could have been many more values. So convert it into smaller products and then find the power and then solve it. Next percentile flowers. What happens in percentage is that the final value is equal to the initial value times the multiplying factor.

[00:17:01]What is a multiplicating factor? So one plus minus percentage change. Good percentage change. So the percentage can increase or it can also decrease.

[00:17:07]So the multiplying factor in case of increase will be greater than one. In case of decrees less than one will be there. This formula. Now this is a universal formula, this percentage profit and loss, this interest is useful in everything.

[00:17:20]After that comes the successive percentage change. Remember that if the same quantity is changing by 2%, then executively. Like one was 100 and got decreased by 10%.

[00:17:29]Further if it increases by 10% then overall increase is decreased, so use minus. If you have increased it then use plus. And put it here. That is x + y + xy / 100, this is for only two. Okay, right?

[00:17:42]Price, consumption, expenditure, three meaning conditions are formed in this. First of all, there should be no change in all three. Okay, right? Correct. Secondly, if there is any meaning in all three, then none of them remains constant. Meaning neither P nor C nor E. Correct? Then we will apply a simple formula.

[00:18:00]P1 C1 / E1 = P2 C2 / E2 If E is constant then your expenditure is constant. So PC equal to constant means price * congestion equal to constant.

[00:18:11]So from here P1 C1 = P2C2 Well second thing is that if the price becomes constant then E / C = constant means E1C1 = E2C2 that's it then if the consumption becomes constant then E / P = constant that is E1 P1 = E2P2 that's it.

[00:18:27]So this is all about the percentage.

[00:18:31]Next comes profit and loss.

[00:18:32]Profit and Loss has very simple formulas. That is SP = CP 1 + profit percentage or loss percentage by 100 What is the relation between SP and MP that 1 - discount percentage by 100 is the discount given on the mark price. Well, if we divide these two, then SP will become SP, so CP = MP times 100 - discount percentage up to 1 plus minus profit percentage or loss percentage. Well, in case of successive discount, the successive percentage change would come to x + y + xy / 100. There will be a minus here. Just keep this in mind.

[00:19:05]Next in case of false weight, true up true measure up faulty measure equal to 100 + g% up 100 + - x% okay g% which is fault due to measure okay x which is if profit is being made in price x tell on what basis in price if profit is being made in price then we can say okay brother let's use plus here and if there is loss then we will use minus okay salary minus saving equal to expenditure you all know this.

[00:19:33]Next comes interest. Now the interest is very interesting. There will be two types of interest in this. Simple interest, compound interest. Calculation of simple interest is very simple. Simple Interest Equation P * R * T / 100 If you do not know this term, then first watch the complete one shot video. After that, look at it again. Okay, right? Those who are able to understand all this PRT, those who have read it, they must be enjoying it a lot. Ok? You must be understanding everything. What is SI? PRT * 100 PRT / 100 Principal Rate and Time The second amount will be Principal plus SI.

[00:20:07]How much will the compound interest be? That is amount equal to principal times 1 + R / 100 power T. If this is being compounded annually, then if it is being compounded half yearly, then we will multiply the time by two and halve the rate.

[00:20:18]In case of quarterly, we will multiply the time by four and the rate by 1/4.

[00:20:23]Good CI - SI for 2 years PR / Square of 100.

[00:20:26]Well CI - SI for 3 years P this will definitely remain complete. What will happen next? So 3 + R / 100 is the difference between compound interest and simple interest for 3 years and difference between compound interest and simple interest for 2 years. So keep all this in mind very well. This complete will remain the same which is for 2 years and along with that in case of 3 years there is three. So 3 + R / 100 come next [sound of clearing throat].

[00:20:52]Time and work. In this you just have to remember a simple formula. That is the efficiency or capacity work upon time. Now the case of pipe and system can also be solved simply with this.

[00:21:01]One comes the Monday Man concept. So what we do in the case of man day concept is that man * day becomes equal to constant. So from here it is understood that when the product is constant then m1 d1 = m2 d2, well the concept of man day hours which is man * day * hour equal to constant, so m1 d1 h1 = m2 d2 h2, right? Like he will tell you that 10 people work for 9 hours every day. He completes any work in 15 days. So if 9 people work for 8 hours every day, then in how many days will they complete the same work? So he will give you one variable to find it, meaning five variables will give you the source of information.

[00:21:43]Ok? That's it. End. Good average mixture and alligation.

[00:21:47]What will we do in the case of average mixture and alligation, what is average? So the sum of all observations divided by the total number of observations. But if the group is formed. Let's assume there are 10 groups. There is some average of some x number i.e. n1 number of students in a group. The average in the second group can be the average of anything. It could be due to his weight. It could be due to his marks. Ok? So it could be about height.

[00:22:09]Okay, right? So there are many such groups g1, g2 up to gn. There are n groups. The average of all is from a1, a2, a3 up to n. Number of students is n1, n2, n3 up to n. So how will you calculate the group average? That is n1 a1 + n2a2 + n3 a3 and up to so on in n by total number of students. That is, add the total number of students in group one, group two and group n.

[00:22:33]That is the group average. Just like that, a case of mixture comes up. A case of mixture and alligation is that you have two quantities. Will mix it. Meaning, the concentration of one C1 and the concentration of another C2 can either give the ratio of mass or the ratio of volume. So, C1 and C2 are the final concentrations. When both are mixed, C comes out, so in such a way that whatever C will be there, it will lie between C1 and C2.

[00:22:59]Okay, right? So let's say C1 is small, C2 is big. What is C1? is small, C2 is large. So C will bring it somewhere here. So you do this criss-cross and get big minus small and big minus small. Do it this way.

[00:23:10]And if you equate the ratio of these two, a formula will be generated. That is c = cm1 + cn2 / n + n. Well, when there is a case of replacement, we use the multiplication factor. Ok? I had told that too. Next comes speed, time and distance. What is speed? There is distance/time. After that, in the train problem, all you have to remember is that the time is LT + L not / ST + - S not. lt length of train l not is the length of object s t speed of train s not speed of object what all can our object be it can be a pole, a man, a train, a station, a man, then at that time take the l not value as zero or if a pole is standing and a train is crossing it then what will be the value of l not, it will be zero well if the man is standing, it is at a stop, the station is not moving, it is stationed, then in that case the value of s not also becomes zero so if the train is moving, two trains are moving with each other, then the object is a train, if the train is moving then s not will not be zero but if the man is standing at the station or if we talk with respect to the pole, then all that is stationary, then take s not as zero in that case you will remember these things in the case of a river boat, two things upstream downstream, after that speed equals to distance/time then what happens in the case of a circular track Well, in the case of upstream, the velocity of the boat minus the velocity of the river, in the case of downstream, the velocity of the boat plus the velocity of the river, remember that the water which helps the boat to move forward in the case of downstream, is clear. After that, if there are two people rotating in a simple circular track, then the one who is rotating from there, the first place where they meet apart from the starting point is the TF.

[00:24:49]What is TF? That the first meeting apart from the starting point. And what is TS? The meeting that will take place at the starting point and what is Neeta?

[00:24:57]This is called Neeta. That is the TS/TF.

[00:24:59]Clear, right? Meeting of starting point means time divided by when it met somewhere in the middle.

[00:25:03]That is what Nita is what? The total number of meetings. These are the three things that need to be found.

[00:25:08]Okay, so when will you use the TF d/a A + B plus sign? When moving in the opposite direction. And the minus sign when moving in the same direction.

[00:25:17]Okay, right? Meaning that two people started moving in the same direction. What is TS? That is the LCM of D / A D / B. Well what is D? So length of circular track. And what is AB? So the speed of this person means there will be two people.

[00:25:31]Okay, right? If A is there then his speed is A, if B is there then his speed is B, well here next if there are more than two people then take reference of one of them. Let's assume there are three people. So please take the reference of the first one.

[00:25:42]A is a reference. What will we do now? So take out the TF. That is the LCM of D / B - A, D / C - A. Right?

[00:25:50]Only two terms will come here. If there are three people, then more than two. So A was taken as reference.

[00:25:54]Remember d / b - a d / c - a. Well, what a is is a reference.

[00:26:00]Took reference from some person. What happens to ts after that? That is the LCM of here as it was done here d/a d/b as many people as there are.

[00:26:06]So here d / a d / b d / c will come.

[00:26:09]t and Neeta what will happen? That is ts / tf that is the total number of meat.

[00:26:14]Next comes P and C.

[00:26:16]In case of permutation and combination, simply you have to remember that permutation is the selection plus arrangement that is npr n / n - rro.

[00:26:24]In case of combination, there is a simple relation called selection that is ncr which is = norial / n - rro * rro.

[00:26:32]Well, now one more thing to keep in mind is that npr = ncr sorry npr = ncr npr = ncr * rorial, both of these are equal to each other.

[00:26:45]Factorial means product from one to n if written as factorial. That's it. In the example, if 5orial is written then 1 * 2 * 3 * 4 * 5 that is 120. Next comes the number of arrangements of n items of which p of one type p of another type and the rest are distinct. From the let's that R number of distinct. That is, its formula will be that is norial / porial qorial rho, it has been explained very well in the class.

[00:27:09]Number of permutations of nth objects where each can be used any number of times is equal to n raised to the power r. I remember the monkey problem, monkey and master problem. Okay, right?

[00:27:22]Well, number of ways of selecting one one one [sound of clearing throat] one of one and more items one and more items from n items that is nc1 + nc2 + nc3 up to so on ncn = 2 to the power n - 1 this was also explained very well in the class.

[00:27:41]Remember this. Ok? Meaning, you should remember all these things before going to the examination.

[00:27:46]Then you should appear in the CSAT exam. Well dividing p + q + r things * p things q things and r things = p + q + r o / p orial q orial r o rial There are two things to remember in the case of circular permutation.

[00:28:02]Firstly, if it becomes clockwise and anticlockwise then for example, if we assume that we have to do permutation of n things which is equal to n - 1sorial, we used to do it by reducing n - 1sorial by one.

[00:28:14]What is good wear clockwise anticlockwise? It is different. But if it is the same then n - 1sorial / 2 comes in probability. So here also we have to remember three things.

[00:28:24]That is, the probability equals the number of favorable outcomes divided by the total number of outcomes or the number of favorable events divided by the total number of events. Well, P of A B equals two means, remember its simple meaning.

[00:28:38]If P of AB is written then P of A intersection B ah / P of B well if these are independent events then P of A intersection B = P of A * P of B. Saying independent means that one is not dependent on the other. Last slide for today's session has some fluff of Venn diagrams and dice. Please take care of this very well. If I have two objects then N A Union B which is called union and intersection.

[00:29:03]Okay, right? This is intersection and this is union. The u written means union written in reverse so intersection so n a un b = n a + n b - times n a intersection b and n a u b u c = n a + n b + n c - 2 at a times means n a un a intersection b yes a intersection b a intersection c and minus a intersection b intersection c and plus the intersection of all three that is a intersection b intersection option c.

[00:29:35]Next let me talk about cube and dice that if I have a cube then I have to make cuts in it.

[00:29:39]How many cubes will there be? So the total number of 1 unit cubes is n cubes. Okay, right? If the cube is of five units, how many cubes will be there in each unit? So 5 cubed that is 125. Okay, what will be the total number of cuts? That is 3 times 5 - 1 if we assume 5 is cubed. So 5 - 1 just arrived. This is the number of cuts to be made. Well, one face colored. Meaning, let's assume that I have colored everyone. How many faces are there in total? In the cube? Six. I colored everything.

[00:30:04]So how many faces will be single coloured?

[00:30:06]That is 6 times the whole square of n - 2.

[00:30:08]How much will the two-faced one cost? That's 12 times n - 2 and what's the three-faced one? So that's going to be eight cubes on the corners. Next comes at least one face coloured. So, it has been said one at least one, right? Meaning that it can be of one face, two face or even three face. So what was One Face's Kaa up to? 6n - 2² to face means 12, if you remember this then everything else will be done. Ok? So 6 times n - 2 squared is 12 times n - 2 and plus 8. Well, no colored face. No colored face. That means how many cubes will be inside the b? So the whole square of n - 2 is five units so the whole square of 5 - 2 that is three sorry the whole cube of 5 - 2 that is 3 that is 27 well at least two faces at least means that there must be at least two faces.

[00:30:56]So maximum three faces because it is the corner one whose all three faces are coloured. Two fuzz, three fuzz. Okay, right? 12 for Two Face, n - 2 and at that end for Three Face. Almost to face colored. Almost to the face means 0 1 2, so what will be the zero one? Look, that is n - 2.

[00:31:15]Where did Zero Face Colored go? Yes. No Face Colored that is n - 2 so No Face Colored it is. One face 6 times of 8 n - 2 and two face 12 times of n - 2 that's it.

[00:31:28]Or you can directly say that Sir, total cube minus that cube whose three faces are coloured, then the total is n, total is n and whose three faces are coloured, that is 8, so n - 8, just use your brain a little and it will be done. Ok? So these are the formulas which you have to memorize before the exam.

[00:31:49]Ok? Watch it two-three times and it will get imprinted in your mind. Ok? So thank you so much from my side and all the best for your exam. Ok? Take care. Bye-bye.