hey friends my name is zee and you're watching me mr easy and welcome to a new video for igcc app matt and today we have rules and examples for permutations and combinations or permits and comments and we'll look into the questions in that session but here's rules an example starting with some basics and before you get into it don't forget to leave a like subscribe and when you're not friendship so you don't miss out on any future videos and we'll get right into some basics so here we have three basic counting principles right here it's number one we have m times n so like basically these counting principles are a way to like find the possibilities or like how to count different things so that m times n is a way to find a configuration of something so when to use m times n is that when you have like this example right here six men and five women in a mixed double competition how many pairs are possible you would just do six times five because it's it's like the basic counting principle and six times five is um it's like a way to find how many service pairs are available so that's on one and we don't typically use it but it's still something that we would use and number two we have a to the power of b and basically a is how many ways to do something and the power the b is harmony of something so for example there are four spaces to fill and five choice for each and the fundamental the fundamental content principle gives us basically how to find it or how to solve this question is you would do five to the power four because it says there are four spaces a field so like there are four and therefore something to fill one two three four and there's five choice for each so there's five ways to do something so there's five ways to fill this five ways to fill this five with the filters and five ways to fill this and you time them all together which is basically five to the power four which is oops four no four four which is right here 5 to the power 4 and 5 to the power 4 right here and number 3 we have n to n factorial and we'll look into this later the factorial sign but basically n factorial is used when trying to arrange different things without any condition select how many ways to arrange five people in a row it'll be five factorial because what this is is that let's say we have five um five seats one two three four five let's say you have five people right so in the first seat you can arrange five people because there's originally five people so five people can never be communicated here times by the second seat since one person has already sat in the first seat the second seat can only be set by can only be set by two uh by four people so times by four and the third seat since the first two seat has been taken the thirsty can only have three people three so the same logic times two times one and this is basically a factorial which is five factorial and we look into what factorial means so we have n factorial where n inside just like a like a number like it could be like 5 factorial 6 factorial it's like a constant so what factorial means is that when we have like n factorial it'll be equal to n times n minus 1 times n minus 2 times n minus 3 but indefinitely to the n minus something like that say a so what this means is that if we have 5 factorial it'll be 5 times five minus one times five minus two times five minus three until it reaches like five minus four because you can't have zero because if it times everything by zero it'll be zero so you can't reach 5-5 so this could be this means that a has to be bigger than n and it's not equal to so as a more straightforward way to know this is that when you have five factorial you basically times basically i'm multiplying the number five by descending order to one so it'll be that five times four times three times two times one and eight factorial will be eight times seven times six times five times four times three times two times one and using this logic we can also know that eight factorial is equal to 8 times 7 times 6 times 5 factorial because 5 factorial is oops because 5 factorial is this part right here this part and basically this is what i meant from here so one thing to note is that zero factorial is equal to one and there's some proof for that but we won't get into uh those proof right now but you can just search on youtube for the proof that zero factorial is equal to one then we have permutations and combinations and for permutations it's basically written as n p and r where n is a superscript and r is a subscript so it could be like five p three like this and permutation is when the order matters order matters and combinations is when order doesn't matter so what i mean is that when you when we have let's say q and r let's say they're both like different people q and r when they're lining up on like a queue for permutation q and r will not be the same as r and q so the first person being in front of the second person is not the same as the second person being in front of the flat of the like the first person however in combinations the order doesn't matter that means that q and r qr is equal to r and q so the first person in front of the second person is the same as the second person in front of the first person and they're both the same and for both permutations and combinations one key thing to note is that the subscript the r has to be smaller than or equal to n because let's say five p four it's correct it's true five p five it's two but five p six the r is bigger than the n so it's incorrect i mean you can't get an answer for that so for permanent the permutations the equation is basically n factorial over n minus r factorial as mentioned so it could be let's say if it's 5 p 3 it'll be n is specifically the subscript so it's the superscript five factorial over five minus three factorial and you could simplify that and furthermore there's a there's a button in your calculator that you can use to calculate the um the permanent the permutations and combinations so i i have the casio like the casio silver calculator with me so if you look at the right hand side above the times and divide button there's there's npr and ncr and if you put if you type shift then times you can put the times button you'll get a p so it depends what calculator you have and you have to try it around but anyways for combinations the equation is quite similar but there's something different this as uh r factorial specifically the equation for combination is n factorial over n minus r factorial r factorial so let's say it's five c three it'll be five factorial over five minus three factorial times by three factorial like so oops like so factor and now we move on to some more basis so here's this diagram to show you the whole like combinations and combinations without restrictions and so let's say we have net in your permutation order and combination unordered you can see the different ways to do it and now we move on to some examples number one calculate eight p five so i could just type into my calculator but i'll just write the whole thing out for this video so eight p five would be eight over eight minus five factorial 8 factorial here so 8 factorial that means it will be 8 factorial times 3 factorial and if you if you cancel out the three factorial with the top factorial you're left with 8 times 7 times six times five times four and so because you cancel out the three factorial so if you type into a calculate eight times seven times six times five times four you get the answer which is six thousand seven hundred and twenty so it'll be six thousand seven hundred and twenty so question number two we have four boys and five girls how many ways to arrange them when the first two have to be girls so let's just show like a queue like a different lines to represent all of the members so in total there will be nine people so like one two three four five six seven eight nine so the first two have to be girls so let's just draw like a box within the first two like so so the number of girls present in the queue there's five girls and the first two have to be girls so we have to somehow put five girls into two goals right here like you find the find like the the the permutation or the combination so the first choice is that you can put five girls right here and after putting five girls you have four goals left so you do the times by four girls and that's it because and there's basically five pd4 so five p4 five times four and once you've like put the two the two possibility for like the first two seats you can just rearrange the rest by themselves so once you have taken two goals in the first two seats there will be three goals left three goals there will be four boys plus three girls which is seven people in total and from the basic hunting principles one of the counting principles is seventh like n factorial when you use it when you're trying to arrange people on like a line or the q and in this case it's the same so it'll be seven factorial and since seven factorial means that you're rearranging seven people on seven seats like for seven different seats it will be appropriate in this case so it'll be five times four times seven factorial and if you put into a calculator you'll get us hundred thousand and eight hundred right here then we have number two example number three sorry so the word history how many ways are there to arrange the word so let's look at how many words there are in history the word there's seven words so basically the the amount of ways to arrange the word will be seven factorial which is equal to five thousand and fourteen right right so number two arrange when o and r are together so let's just use examples just now so let's say the seven uh seven words it'll be seven blocks one two three four five six seven oops and seven so o and i have to be together so let's just put group them together o and r let's just put o and r and since o and r can rearrange that by themselves so you do two factorial because it could be o r or r o and it would really matter because ro is not the same as oi in terms of like the whole word and whole meaning and because o and i have to put together and it's not that it has to be in the front or in a certain position that means that you have to like we have to like arrange it along with other uh other letters so what i mean is that or could be in this case it could be like this or it could be like this it could be in this position right here or it could be in this position right here so you don't know which element which which position it is in so you have to permute or like you have to rearrange it with other letters so just count how many blocks there are left will be one two three four five and all r is all are together they combine together to form a big block so it'll be the sixth clock because you can't separate o and r together and remember from the basic counting principle to arrange six people or like six blocks on the line you do six factorial so basically you'll have two times by six factorial so it'll be two factorial times six factorial and if you put into the calculator two factorial which is two times six factorial we get one thousand four hundred and fourteen so one four four zero right here and number three where o and r are not together so what you could do is that you could like just zero one two three four five six seven you could find like all the ways when o and r are not together it could be o r o r r are different like o r blah blah blah but it would take way too long so one really nice treat is that you can use the total number the total arrangement for the letter which is part a five thousand and fourteen minus the amount of times when o and r are together which is part b 1440 and if you if you have a total number of arrangements take away the times when they are together that means that you can find the times when they are not together which is what you're trying to find which is equal to 3 600 and it's a nice trick to know and then we have some more examples how many ways three prefects can be appointed from 10 candidates so what you can do is that you can just think of if like let's say if you have three candidates right here three would a b and c be the same as b a and c and the answer is yes because there are still three that are elected and they are the same like um same people if you just rearrange them around doesn't really matter the arrangement so in this case it's combination of bc so to appoint three candidates from 10 candidates you'll be 10 c3 and if you put into your calculator or if you simplify it you'll get a value of hundred and twenty and we'll move on to the next question i won 14 inches and and two polo shirts how many ways to arrange if i have 10 sorry it's 10 t-shirts and four polo shirts so let's just think of it this way we want four teachers and two producers let's just take the polishes for example the polo the polo let's say we're near polo a and polo be selected we'll pull out a and polar beans being selected the same as polo b and polo a being selected and it is the same because you still have both of the t-shirt and the order doesn't really matter when you pick it up because you still have both of those t apollo so in this case it will be combinations as well so when we have 10 t-shirts and we want 14 shirts you do 10 c4 times by because they are the same like they happen to be in the same group because it's not like different elements they have to be in the same because you're trying to find how they like reliability your relationship affect each other and there'll be four protocols so b4 c2 and if you put into a calculator 10 c 4 oops let's just put 10 c4 it'll be 210 and 210 times 4 c2 it will get us 1260 that's the answer and now we move on to a new question so four of the lessons in mexico are selected fine when there's no restriction so it says so this question basically just asks if it's selected not like if it's like arranged in the line semester club has six words so it's six letters so four are selected so if you were to select four would m e x i be the same as e m x i if you want to just select them so that you have the letters and it will be the same because you will still have these four letters if you take that in order of m e x i and in order of e m x i and it's not like they have to rearrange in like a like a letter form or like in a line in this case so it just what it just have to be selected selected and they're the same so it'll just be 6 c4 c4 which is if you put into a calculator it'll be 15. right so number 2 m has to be selected so from the four letters right here m is reserved and so just put the first block for m and then just don't look at it so there are five letters left and five letters can be selected and put into three slots so it'll be five c three and just select it just to be sure selected and therefore the answer is just five c three and if you put into a calculator five c three oops five c three will just get us ten like so then lastly we have the last question a committee of five members to be selected from six seniors and four juniors fine if there's no restriction so to to know that there's no restriction we know that we can just combine seniors and juniors together to become 10 people in total and we know that the order doesn't matter because if you have let's say person a and person b being selected it'll be the same as person b and person a being selected so it'll be 10 c5 which get us 252. then we have b has three seniors so let's let's just draw the line out again one two three four five like so so three has to be senior so let's just preserve this for seniors and this is for juniors so there are six seniors in total so that means we have six in total and they have to be fitted into three different slots so it would be six c three because person a and person b and person c will be the same as person b a and c this will be times by the junior they are four juniors and it can be fitted into two different slots times four c2 so the answer would be six c3 times four c2 and if you put into the calculator six c3 times four c2 you get 120 all right so 110 and c has at least one junior so you can just do the way where you have like ef one junior plus two junior plus three junior plus four junior but in the other way you can just find when you can just use the truth just now find the total number of like um the combination or like no restriction which is 252 from the start minus when there's no junior so let's find when there's no junior one two three four five so when there's no junior there'll be there'll just be six seniors so six c five so 255 sorry 252 minus six c5 and then um six c5 is basically six so 252 minus six and that will give us 246. and that's the final answer and that's it for this rules and examples for permutation and combinations and this video is really confusing because when i first learned this concept it's really confusing for me so what i did was i just did questions and examples and answers and look at the answers to check if i'm correct and to use the right principle for each question so if you have any question or any like confusion just drop it in the comment section and don't forget to leave a like subscribe and bring the information back so that you don't miss out on any future videos and as i mentioned if you and if any comments or contracting feedback on my channel on my youtube or my website just chip that in the comment section and i'll reply to them and you can also email me if you have any questions or any questions that you want me to do and i'll reply to you and also check my social medias in the description for example linkedin or instagram for more daily content and if you need any learning resources or any teaching resources just check it out in my website or you can type it up on your browser at www.emsee.com and i hope you'll find it useful and helpful and i'll see you all in the next video which will be questions for publications and combinations but until then stay safe and happy learning [Music]