GCSE Maths Tuition Lesson 21 ALGEBRAIC PROOFS EXAM QUESTIONS

okay so as I get started so yeah algebraic proofs um they come up somewhat frequently maybe every other exam series or so um and also the product rule for counting can come up uh a decent amount so what I'm thinking of doing is we'll just go through a bunch of the different proofs they can give you you guys are going to catch on pretty quickly because it's a very similar method each time just for the different types of proofs protocol for counting again there's a few different scenarios I'm going to explain it to you and then just tell you a way on how to kind of remember it as well but it can help to know exactly how it's done so you can apply it to different scenarios and things like that and then at the end if we get time you can actually use it to work out you know what the likelihood of winning the lottery is which I think is somewhat interesting uh maybe you guys have been interested in that as well if we have time but yeah I think we can just get started uh you may have done these before but some schools don't actually do this topic because it doesn't come up that frequently and they don't have enough time so I just want to make sure that you guys have some exposure to it so it says prove algebraically there's some of any two consecutive integers is always an odd number so yeah if you've got the right idea so the first things first is consecutive means they have to be one after another so for example if I write out right these two architective these two are consecutive these two are consecutive it just means they're next to each other in a list so if I just say an integer is n right then the next integer which is consecutive would just be n plus one which hopefully should make uh sense and that would just be two n plus one right which is exactly what EV said but hopefully it's clear why this would be an odd number if you take any number and times it by 2 which is the first part of this uh calculation 2N you always get an even number and then when you add a 1 if you add one to any even number you always get an odd number so that's why it's an odd number okay because again this is always even an even number plus one is always going to be odd so in starting with something relatively soft but there are some pretty tricky well some what tricky ones okay and I'm just gonna keep going but as soon as you see something you don't understand then let me know but again there is a chat delay so keep that mind prove algebraically they're the sum of any three consecutive even integers is always a mem is always a multiple of six so an even number can just be written as 2 times n because 2 times any number is an even number now the next even number if I write out the even numbers here these are the consecutive even numbers right 2 and 4 consecutive four and six six and eight so what you do is you jump up by two each time so the next one would be 2N Plus 2. and the last one will be 2N plus another two so plus four then all you need to do is add all of these together the 2N plus 2N plus two I will say that whoops for any of you that want to do a level maths proofs comes back in full force a bit more beefy than it is here but uh it will kind of come back again good question Evie so you can't and the reason for that is n is normally just defined as any um any integer okay so if I just say n and it has to be uh even that that wouldn't work and in fact if you subbed it in you'd find out it wouldn't work it has to be 2N because again yeah because you don't want to limit it generally speaking n is just any integer at all okay so 2N would be how you'd Define it and that's how a lot of these um proofs work but good question if I gather the like terms I have 2N plus 2N plus 2N and that's 6n and then I have 2 plus 4 which is 6. now he's saying show that it's always a multiple of six now this is a very common part of the proof so first of all you have this right you need to show that it equals 2N plus one right that's one thing so prove it's an odd number to prove something is a multiple of blank all you have to do is show that you can divide it by that number okay and the way you do that is by factorizing it out so equals I could write this as six n plus one okay and that is my proof at the end because I can obviously divide this by six because I have a six out front if I divided by six it would just get rid of the number in front but if they said multiple of 12 or 3 all you're doing is pulling out or factorizing out a 12 or 3. and now if it said prove it's an even number then all you're doing is factorizing out a 2 because even numbers are multiples of two so so far there's only two variations you have prove it's odd where you're going to have to make it equal to n plus one or it's going to be a multiple of a certain number or even where you just pull out whatever that multiple is so it could be a multiple of 3 6 12 whatever and if it's an even number you're just pulling out a 2. which hopefully should make sense are there any questions about that at all again we've only done two examples so if it doesn't make full sense right now that's okay okay good okay so perfect so again in this question this thing here is different but it says prove it's a multiple of 12. so we're going to do is fact factorize out of 12 and we're done that's it so if I were to expand this 3n plus 1 squared would give me 9 N squared plus 6n Plus 1. if I expanded that then we have minus and if I expanded this bracket I'll get again 9 N squared Plus 6n plus one uh hang on I've made a mistake that should be a minus in please there we are minus six seven okay now we expand the brackets one more time so can you see that the 9N squared minus 9N squared goes away the plus one the minus one goes away so that equals 12 n so obviously that would be 12 times n so it must be a multiple of 12. right it's 12 times a whole number which is always going to be a multiple of 12 right it could be 1 2 3 4 Etc so again it's the exact same thing as this so there's a few other styles we're going to do and there are some exam questions where'd you get minus one from ah good question so in this last step here I have to expand this bracket and as you can see there's a minus in front of it and since there's a minus in front of it you need to put a minus in front of everything inside the bracket so you do minus times 9N squared so that's minus 9N squared minus times minus 6 n would be plus 6n and then minus 1 times plus one which is minus one if you'd like it's just flipping all of the signs inside um are your papers out yet again only one a at the moment I know it's kind of frustrating um my friend who needs to do the mark schemes she had she was very busy recently so she hasn't been able to write them up yet but I'm hoping they'll be done relatively soon if not I I can just write them out but she has some very nice handwriting okay now here's a slightly different proof so so far again there's two types right prove it's an odd number so you want to make it two n plus one prove it in multiple of a certain number including an even number and then that's that the next type is you need to prove that something is a square number now I'm going to show you how you do this but it's really not too bad but it says prove algebraically that the sum of N N plus 1 and N plus 1 is always a square number so sum just means add so I'm literally just doing n plus 1 add if you want to put it in Brackets n plus one so now I expand the brackets right N squared plus n plus n plus one right so again n times n is N squared n times plus one is plus one as plus n sorry is product multiplying product is multiplying yeah and then plus n plus one cool now if I simplify this I get N squared plus 2N plus 1. so what you need to do in this case if you want to prove it's a square number you're always going to be factorizing what you have okay so whatever it is you're going to factorize it because to show that something's a square number you need to show that you can square root it okay now I can't really show that I can square root this that's pretty tricky right so instead I'm going to factorize it so I'm looking for two numbers at times to one and add to two while one and one come to mind right so then I have M plus one n plus one and the way I write that is n plus 1 squared if I can write something as a thing with a square it means it must be a square number right 2 squared is a square number it's four three squared is a square number 4 squared is it has to be a square number right because I'm squaring it so it has to be a square number after I'm done squaring it I said the word square like kind of lost all meaning but I hope that makes a bit of sense if it doesn't then please let me know this is the other type that they can give you so there's three types so far prove it's odd prove it's a multiple and prove it square and by the way when I say prove it's multiple that counts prove it's an even number because even numbers are just multiples of two okay so the method is exactly the same three ways so far okay I think that's okay hope that is anyway okay perfect so again we have yeah prove that this is a multiple of 12 for all positive integer values I'm going to start picking up the pace ever so slightly so we can get onto a lot of stuff because I've got a ton of questions including exam questions and things like that so again you're going to expand both of these brackets so two instance 2m will give me four N squared plus so 6n plus 6n would be 12 n plus nine so I've done this very quickly so please let me know if you aren't sure 2N plus 2m will give me 4N squared then we have minus six n minus 6n so we have minus 12 n then we have Plus 9. it's again I'm going through that slightly quickly just so we can uh get through a bit more but please let me know if it's confusing and then again we want to expand the brackets one last time so 4N squared plus twelve n plus nine I would recommend doing this intermediate step because it's very easy to mix up the signs in an exam okay so again minus four N squared an introduce times out so minus one times four n will give me minus four N squared minus times minus 12 n would give me plus 12 n minus 1 times plus 9 I'm giving minus 9. so I'd literally recommend doing that so you don't make a mistake in the exam the plus 9 and the minus 9 cancel of course the plus 4N squared cancels with the minus 4 and squared of course so we're left with 12 n plus 12 n which is 24 n and if you remember we need to prove it's a multiple of 12. so we need to show we can factorize out 12. so we have 12 times 2N which means it is a multiple of 12. perfect just um for your information so again if you're doing a levels this comes back but the reason why we do it with algebra is because if you think about it if we try to prove this by subbing in numbers we would be here until the end of time because there's an infinite number of integers so if you do it algebraically it proves it for every single number in existence that's an integer that's why it's very useful because otherwise you have to just do like a billion different cases or an infinite number of cases very good okay okay so again always a square number so I'll do this one because this is like the second square number one we've had so again it says sum which means add and as um Lila asked earlier product would means you type would mean you times them and they can ask for product so that's a really good question so we have n plus two n plus one Plus n plus two in Brackets so let's expand the brackets n times n is N squared plus 2N plus n would give me plus 3n and then plus two plus n plus two so N squared plus 4N is four this one's gonna be looking for so again we need to factorize it to show that it's a square number and it needs to times to four and add to four so the only numbers I would work are two and two so we have n plus two n plus two again I am doing it in like a a slightly longer way in the sense where I'm writing everything out in full but that's just to make sure everyone's fully on board you can go from this line to this end line here that's fine yeah you get n plus 2 squared because I'm squaring it it means it must be a square number again once it gets too easy just let me know and we can go ahead this one will be good to do as well too easy so far okay it'll get slightly more challenging slightly more this one is very similar to the consecutive even numbers one so if you want to ever write an odd number you would write it like this 2N plus one so I'm not sure how many people have just recently joined but just as a reminder if you times any number by two it becomes even if it's an integer if you add one it has to become odd so if you ever want to write an odd number you always write 2N plus one now to get between all of the odd numbers to get to the next one you're always adding two right so to for the next uh consecutive odd number it'll be 2N plus 3 and then 2N plus five because again we're just adding two each time to get to the next one it wouldn't be two n plus two because that would be an even number and all we're doing is we're adding all of those up so 2m plus one is two n Plus 3. 2N Plus 5. so we get two plus two plus two is six n one plus three is four plus five is nine cool and prove it's a multiple of three which means we factorize out a three so again three and we get two n plus three which means it must be a multiple of three I hope that's uh okay so far there are some other types which might get a bit bit tricky a linear sequence instead of consecutives um I suppose there's no reason why not but I've never seen it so if you mean like they could give you prove that n plus two plus n plus five plus n plus 10 or something Maybe but I've never seen it consecutive even numbers will be the same thing as the odd numbers except you're just going to be adding 2N two n plus 2 and 2N Plus 4. we did do a bit earlier but again all you're doing is you're doing 2N plus 2N plus two plus two n plus four and then you'd factorize out a six so let's do that I think that should be okay right this can get this is a bit more interesting uh these next two so if you look here prove algebraically that the sum of the squares of any two even positive integers is always a multiple of four so if you notice it doesn't say consecutive right it just says any two even integers okay so my my first integer would just be 2N because if I want to write an integer I can just write 2N now if you want another integer which could be any number you can just use another letter okay so it we might use 2m but technically speaking you can use any letter you like you could say to q2z to capital N if you really want to be a agent of chaos anything you like but is it okay if I just use two n and two M because they're next to each other and you see how that shows that they're two different numbers but they're not necessarily next to each other they could be any number okay it's getting slightly more interesting perfect okay and then it says prove that the sum of the squares so with this wording it can be a bit tricky sometimes but when you take the sum of the squares you're squaring first then adding okay so it's not this it's not that instead it's 2N squared plus 2 m squared like you're doing Pythagoras the sum of the squares it's not you sum them then Square them you square them then sum them and it will give you two completely different answers so if I expand that I get 4 N squared plus 4 N squared and it says proved there's always a multiple of four well again I can just factorize out a 4 and I get N squared plus M squared so what I'm hoping is that this isn't tricky mathematically but what is tricky is that you haven't seen these situations before but once I give you like the hint like for example choosing n and M for example you might not think about that but once you have it I think it becomes somewhat easy after that right but it's the thinking about the beginning that's the difficult bit thinking about the beginning yeah okay that makes sense that can be slightly tricky but after that I think it's like this math isn't too bad it's just actually thinking about it so I'm hoping to just arm you with the knowledge of which one to choose and then from there you'll be fine so what if it's odd so it's literally the same question but it's odd well again if I choose one odd integer it's two n plus one if I want to choose a completely different one and again they aren't necessarily next to each other you just do two M plus one so would you guys want me to go through this or is this okay again you're squaring them and then adding them and then if it's always even you can literally write this as multiple of two it's okay perfect good to hear okay people are some of the squads of any two consecutive integers I think that'd be okay so in this case you would have n and M plus one Square then add them and then you want to show at the end that it is two n plus one yeah actually this one it's not strictly speaking to n plus one okay so I'm going to go through this one so again some of the squares so all we're doing is doing N squared plus M plus 1 squared it's slightly different that's why I want to go through it but I'll go through it relatively quickly N squared plus 2N Plus 1. so then if I add this up I get 2N squared plus 2N plus one so if you look it doesn't look like 2N plus one right so it's it's not going to be just 2N Plus 1. so we have to do instead is this beginning bit here I can show you that it's always even right and if I show that this beginning bit's always even if I add one to an even number it becomes odd the way I show this is by writing two N squared plus n plus one this bit here is always even because it's Times by two and if I add 1 to an odd number sorry if I add one to an even number it always becomes odd and then you can just write like even plus one equals odd that's why it's three marks there's like a slight bonus bit if you like is that okay the next one's pretty good as well actually maybe I'll see the next one too okay so again it's a bit of a trick but similar thing right yeah this next one I'll do this next one and then hmm and then I'll see and then I'll just ask you how to uh you guys if you're okay I'll do this next one then I'll then I'll leave it to you guys to decide which ones we do so again anyone that's watching you can like basically vote we'll run it like a democracy after this one for now it's a dictatorship so prove that the sum of the squares of two consecutive odd numbers is always two more than a multiple of eight that sounds really complicated right but let's have a look squares we've got some consecutive odd numbers and then two more than multiple a if you break it down it's not too bad right so the first bit we're going to look at is the consecutive odd numbers so again we have two n plus one for what our first odd number and then 2N plus 3 because again we add 2 to get the next odd number right Okay so we've done this bit good now we have to sum and square them so we're going to do uh the sum and square bit now so that's going to be 2N plus 1 squared plus 2m Plus 3. squared also I would like to point out um now just doing this bit here like these beginning bits can get you marks so if you're ever unsure of how to do a proof just do this first bit and generally speaking you can get a couple of marks maybe one maybe two but what you will get is more than zero okay okay now let's work this out so we're going to square and expand it so again I'm going to do the expansion a bit quickly but if you're unsure please let me know I really don't mind and that'd be plus Point squared uh Plus 12n and we're going to gather the like terms so we have 4N squared and 4N squared which gives me a N squared so I'm going to press those out then we have 4N and 12 n so that's 16n then we have Plus 10. okay this is a bit interesting but the trick here is it says it's always two more than a multiple of eight so what I want is I want eight brackets with stuff in it plus two that's my ideal situation right because again two more than a multiple of a I'd have eight times some stuff plus two so what if I just row plus 10 as a plus two I get my plus two and I can factorize out the eight so that's exactly what I'm going to do spotting that you know you know it was I didn't oh God words are hard it didn't take me that long to spot it but again I'm used to these tricks that they play on you but you have to just be open to to this remember that when they say exactly it's really sneaky when they say this prove that the sum of together is this it's definitely true in a level you will prove a statement true or false with GCSE everything they say is true in the context of proofs so I know that it has to be two more than a multiple of a so then thinking and I did I did this in my head but again I've done proofs for years longer than you have and I did a level proofs and University proofs and stuff so take it with a grain of salt but this is kind of what my thinking was okay so it does test your imagination quite a lot and I don't know how to imagine imagine it if you guys are I'm not imaginative at all but I have the benefit of like five years experience on top of you guys probably more than five actually actually I think it's closer to ten which is depressing but yeah hopefully that uh makes it a bit makes more sense this sneaky but that's why I wanted to show you this one because it is sneaky okay we're making good time we've done 12 questions in in 27 minutes it's one every two minutes not bad um this one here again it's a bit longer but what I'll do is I might not necessarily again you guys can decide if we go through it but it says the difference which means you're going to subtract right between the squares of two consecutive integers so all you're doing is you're taking n and n plus one is equal to the sum of those integers so what you do is you do n plus n plus 1 which is 2N plus one and then you'd Square these and subtract them and show it's the same thing that's it again I don't mind going through it but honestly if you just square and subtract these you will find that it equals this there's no tricks there's no tomfoolery no mischief it's pretty much what it says on the tin and again you would do n plus 1 squared minus n right because you take the bigger thing minus the smaller thing because when you say difference you don't care about negatives what you're just doing is finding what the difference is between them so you do the bigger minus the smaller is that okay God okay seems like Evie's the only one that voted so she wins that democratic election for that one well good yeah perfect okay lighted it as well this one again there's no tricks there's no Mischief here yeah it's a rigged election exactly I was like Robert Mugabe guy right in Zimbabwe anyways um pretty much apparently the sums of the squares of two consecutive even numbers is always four more than multiple eight it's not like the tricky one but from before you literally get eight n square plus a ten plus four you just factorize the eight N squared and a n to a to that it there's no trickery here no three marks not bad you don't want to see that's okay and now we have the uh exam questions disappointing three marks three marks to three marks but the reason being is because it doesn't come up that often they don't expect kids to have say kids students to have um revised it and schools have covered it but the thing is that's an easy three marks right like you know that's a steal in my opinion it's quite um quite good in that sense this has gone to the exam questions four marks so prove that and they give you a fairly long expression is a square number so what a lot of students so I actually went through this today with some kids at school some students keep calling kids not kids okay at school today and they saw this and they panicked and they said ah right that's the literal reaction was ugh but the thing is it's longer but it's worth four marks it's not bad if we expand this in steps it won't be too bad I promise so we have 35 expand the bracket so 3x times 3x is 9x squared 3x plus 3x would give me 6x Plus 1. okay so I've done these two bits uh and then you have minus 2X times 4X which will give me minus 8x squared then we have minus 2x times minus 3 which gives you plus six x so let me go through the uh same process 9x squared minus 8x squared gives me x squared so I'm going to cross that off now 6X plus 6X is 12x what's that off and lastly 35 Plus 1 is 36. now okay it times to 36 and adds to 12. so if you can't spot this immediately keep in mind we can use our brains and okay that sounds really sarcastic I don't mean it sarcastically but we know it's going to be a square number 36 is a square number so I'm going to start with what about six and six would that work it does 6 and 6 is 12. yeah perfect so if you can't spot it immediately you can use that as a bit of a bit of a hint Right Use the question to defeat the question and that'd be X plus 6 squared that's four marks you get a mark for expanding Mark for simplifying and your mark here and a Mark here done four marks and you could you guys could probably do that in about a minute I went pretty slowly because obviously I'm trying to teach you guys it cool 30 seconds yeah 25 on a good day the next question I found it was actually somewhat interesting I say I you know I say interesting you know it's GCSE amount it's not exactly interesting but check this out they give you an expression for a different type of number technically speaking you are meant to be able to recognize triangulum triangular numbers but this is the expression for the consecutive triangular numbers okay and it says prove that the sum of two consecutive triangular numbers is always a square number so again slightly more interesting slightly so we need to do is add these so let's do that n plus one over 2 class M plus 1 M plus 2 over 2. now it's a fraction so I can just add these straight away but I could also expand first it's kind of up to you which way you want to do it um I'd rather combine the fractions I think I'm 100 sure what I'd want to do actually oh perfect straight line and now we expand n times n is N squared n times 1 is n I don't know why it took me a second to think of that n times n is N squared again so again I'm doing the second set of brackets now by the way and then you have n times 2 which is two n then n times one is one n so that would be 3n overall and then one times two which is two over two gather all the like terms N squared plus N squared is 2N squared so let's cross those off n plus 3n is 4N and plus 2 on the end over two and it's our lucky day the two actually cancels with everything in the top so what we can do is divide everything by two so we're left with N squared plus 2N d That's one and we're trying to prove it's a square number now you might remember from one of the earlier questions this is just n plus 1 squared but if you're not sure times is to 1 adds to two so that would be one and one so you have n plus one n plus one I'm very well thank you how are you AMC so yeah that proves this a square number so again four marks so they try and throw you off by giving you a different situation right you've never seen the expression for consecutive triangular numbers but I didn't really care about it right I don't care it's triangular numbers they could have called it whatever the hell they wanted to and still get the same result okay so this here would be remember how I said 10 of your marks are unseen where they throw something new at you this is an unseen part right so that kind of actually makes the rest of the paper easier if you think about it but again they could give you some other type of number they could give you a cylindrical numbers or something and give you a new definition but it's the same process I think that's okay so oh well done thank you for saying you enjoy my videos yeah best of luck on your channel too okay um so that's it for proofs I think that's a pretty solid foundation the next thing we want to go through is it's called the product rule for counting sounds a bit fancy but all it's about is finding the total number of combinations right so let's say if I have I go to a restaurant there's 10 starters and there's 10 main courses or sorry 10 stars and nine main courses how many total combinations are there how do I work that out that's all it is now it's going to start off easy but then there gets to be some funky business okay so this is kind of an example of a fairly basic question but I'll use it as an explanation of like how this whole thing works so 12 boys and 15 girls in a class one boy and one girl will be selected to represent the class on the student council work on the total number of ways of choosing a boy and a girl now again you might instinctively know the answer but let's go through the boys and the girls if we pick one boy number one how many goals could he be matched with well there's 15 goals he could be matched with right okay what about boy number two he also has 15 possible goals to be matched with boy number three also has 15 number gods and all the way up to boy number 12 who will still have 15 guards he can be matched up with so in other words there are 12 times 15 combinations so I'm hoping that you know Banks sense right which is 180 possible combinations yeah so again this might start off easy I fully I fully accept that but I need to do it this way to make sure you guys understand a bit later on because they can introduce some weird stuff okay there are 17 boys and 14 girls one boy one girl we selected well okay boy number one match with 14 girls boy number two Master 14 go and so on so once again 17 times 14. so hopefully you can actually see why they call it the product rule for accounting right because all we're doing is we're taking the product the multiplying of two numbers and that should be 238 by the way a lot of these questions are actually calculator so don't worry too much about working out but again good old-fashioned column method the next type of question they could give you is something like this now 14 boys and X number of goals in a choir 1.1 goal will be selected Taylor says there are 152 ways of choosing a boy and a goal could Taylor be right so what they're basically saying is that 14x equals 152. okay so means X should be 152 over 14 which again you'd have to use a calculator for this because obviously you can see it 152 doesn't go into 14. 10.857 so Taylor isn't right because it's not an integer right we can't say there are 10.857 girls in a choir right I mean how could you have 0.857 of a person right doesn't really work so it's another way they could kind of throw a cardboard so far I'm assuming this is fairly easy so if I show you the next question we all okay with that and what about the next one after that if there's three combinations does anyone want me to go through any one of those two questions yeah so that's fine Lyla ryuko that AMT anyone else again uh so Lila do you want me to go through one oh no no okay okay cool you're right I did say Johnny wait Evie are you you're okay right I realized I asked you two different questions I said do you want me to go through and then I said are you okay so if you answered yes yeah cool and again here it's the same as this previous question up here right you would do uh five times six times x equals 130 see if x is a whole number done okay again perfectly fine mule deal classic this is fine two this find two same thing now here's where things get slightly more interesting I mean very slightly you can have different situations like this so there are 52 cards in a deck Peter's going to give one card to Casper one to Kelly how many different ways are there of doing this so we give one to Casper first right so he has 52 possible cards he can get now Kelly only has 51 cards she could get because she obviously can't get the card that Casper has right so in this case it would be 52. times 51. do you want to go with that should be 2652 I promise you why 51 okay good so think about it right so if I'm let's say you and I are playing cards right or maybe you and Lino are playing cards if I give one card each to you right once I give you a card there are 52 cards you could get because there are 52 cards in the deck but once I've given you one there are only 51 cards left in the deck so the options that Lila has is only one of 51 cards instead of 52. because one of the cards is taken by Casper see it gets a little more spicy a little more spicy like I said we can actually work out the probability of winning the lottery if you guys like a bit later on no so yeah in this case it's literally in a decks ob1 it's very careful with the wording so yeah it is a bit tricky that's why I wanted to go through a couple of these so let's do 11 right so the 52 cards in the deck same as normal but Angel's going to give one to Ben Chris and Dylan well it's going to be very similar to what we just did right Ben has 52 possible cards to get but then he has one card so now there's only 51 for Chris and then again one lower will be 50. foreign other level so I'm trying to remember the rules for sequences in series or like summations and things like that or geometric arithmetic pretty tricky yeah EV live is that okay that's once again we're going to times that again if any of the numbers don't make sense please let me know cool and then I'm going to times I'm just gonna do answer times 50 because I really don't want to work that it's 132 600. I mean that's how many combinations there could be that's kind of crazy if you think about it right like if I just give us three one card each from a deck there's 132 600 combinations that it could be I guess that's why poker is so exciting for some people yeah that's pretty interesting converters and divergences from a level yeah that's quite cool further math stuff that you're doing okay and here is where it gets fun okay 52 cards in a deck Tommy's going to give two cards to Jay how many different pairs of cards could Jay get so I'm gonna call this J1 for the first card J gets I'm going to call this J2 for the second car um oh cry oh that's horrible for the second card J gets right cool the first card he's going to get there's one of 52 options and the second there's 51 options right so you could do 52 times by 51. which again we've already done before so you might be wondering why I'm finding this so interesting it's two six five two it says how many different pairs problem with this so the problem with this is this actually double counts so what I mean by that is let's take some examples um how well do you actually okay I'm not going to use the fancy suit or anything but let's say Jay gets a 2 first and a three second right that's fine when we do 52 times 51 that actually includes this as a unique combination so a two and a threes and a three and a two are considered two different combinations okay problem with that is if you want the number of different pairs so if they ever use the word different or unique pairs you need to half your answer okay I'll give you a second to digest that while I read um freckles comments so I'm in my first year so it'd probably get worse but stuff like component stuff yeah that's pretty it seems like uni in my country does further math on UK level um to be honest the first year of university for for me personally we covered a lot of a level stuff and like further at a level so um yeah it's it's kind of like it's not a recap I would say but it does kind of make sure you know the a-level stuff it goes a bit further but it is a lot available I finished eight levels in London moved country with the university so that's interesting oh cool I've got semi levels I'm not the brightest don't worry look grades don't matter that much further math topics you have things like um complex numbers uh various types of proof slightly further integration and differentiation than what you would do in a level but yeah so okay either you see uh Lila how is that for you is that okay good so again the rule I'm going to give you is if they ever say different or unique or anything like that or if they say this is going to seem weird at the moment but each team plays each other once anything like that you divide by two after you times it okay because they're looking for they're looking to eliminate double counting we call it right so getting a two and a three is the same as a three and a two if we eliminate that by halting it so again we have something here we have 30 students in a class two students are going to be selected to receive a prize how many different pairs of students could be selected okay so there's 30 students to begin with but then after one person's been picked there are only 29 students left right so we do 30 times 29. which is 870 but if you look it says different pairs okay so again for us we're just going to divide by two because that's the rule okay but again try and think logically right if you call the students a b c d Etc student a and student B being picked is the same as student B and B and student a being picked so we're halfing it but either way it's fine I can go over a question if you have one maybe that's totally fine um there are a couple more that I would like to go through um should I jump straight to the exam questions make sure we have time got my final twice hopefully I'll pass but learning all these rules in a few hours is it possible let alone knowing which shieldable to use I'd advise just doing practice questions instead of trying to memorize the rules it will help you more but best of luck with your final so again if you look here there are 10 teams in a football league two teams are chosen work at the number of different matches so again you're doing ten times nine ninety divided by two 45. two marks okay I think I want to start going through some exam questions a bit about okay these are exam questions but going through some of the recent exam questions that came out which are a bit trickier look at this number of handshakes how exciting right okay here we are this one was a bit tricky uh actually I'll leave it up sorry Liam's trying to remember a three-digit code he knows the rule that the first digit is a cube number the second digit is a factor of 16. thank you very much Freckle I appreciate that and the third digit is an odd number Liam tries at random a code that matches the rule work out the probability okay so if you remember probability is the number of outcomes you want over the total number of outcomes so there's one correct code and we need to divide it by the total number of codes right does that make sense because there's only one code that works hopefully and we need to work out the total number of combinations we have so the trick here is that it's a single digit okay so the first digit is a cube number let's think of all the cube numbers that are single digits well if I go through my Cube numbers one cubed is one two cubed is eight three cubed is 27 so in other words I only have two options because after that they start taking up two digits which is not allowed so I have two combinations or two possibilities for the cube number the second digit is a factor of 16. okay let's list all my factors of 16 well we have uh one and sixteen we have two and eight and we have four and itself right uh there are no others good so the ones that are single digits are one two four and eight so there are four options here actually it's probably awkward that I'm putting a thing here but it looks like a negative sign I'll leave that the third digit is an odd number well let's list our odd numbers we have one three five seven nine eleven so there's only this many odd numbers that are single digits so that's five so the total number of combinations is two because there's two possibilities for this first so I like to actually put it in a box sometimes right there are two possibilities for this first place which can match with any four possibilities here which can match with only five possibilities here so in other words you're going to times them all together and in this case we're not going to divide by 2 because the order matters okay so for example getting uh a one and a 3 here is different from getting a three and a one here right if you think of a password so two times four is eight eight times five should be 40. so the actual probability which is where some people lost marks because they forgot to do it would be one in 40. so you get one mark for this and then you basically get three marks for the working out along here but if you just get the right answer you get all four marks okay but in terms of working out you need to have like some of these steps put in so evil is that okay and others by the way if anyone else is here I can see there's more people that are in here but I guess they're shy which is fine yeah Lila perfect oh yeah I was gonna go through um work out the probability for the lottery I thought that might be interesting are you guys interested in it or should we just jump straight to Evie's question so again let's do democracy there's only two of you so this is going to be interesting if you guys have differing opinions but who would like to work out the probability of winning the lottery just as like a test both of you don't mind okay how about Evie let's do your question because it might take a while who knows right so if you just want to like send the question or whatever from your mock yeah let's hope that I can get it right oh there is another question huh Democratic extreme yeah I do have more questions here there's a character and there's yeah okay please okay there's eight bodies floor faces and five hairstyles what's the probability two characters are exactly the same so if they're exactly the same they would need to have the same body the same face and the same hair right yeah I think I know what they're going for here so there's 160 combinations I'm moving on to check I'm just gonna assume you're right with that so the probability of picking a character is one over 160 right that's the probability that it's the um that's probably the getting of one particular character right so in order to get the exact same character again I'm assuming they they make two unique characters right not unique sorry they make two characters what's probability of them being the same that's the question well it will be Times by one over 160 again wait is that the question where it says what's the probability so if they're making two characters what's the problem see they're the same because if so yeah so then all you do is do one over 160 squared right or one of 160 times one over 160. one over twenty five thousand six hundred no not add so