foreign [Music] [Music] okay so in this video we're going to have a look at the product rule for counting now the big thing with this topic is it's relatively simple actually to go about actually doing but the thing with this topic is some of the language as it changes throughout the course of the topic so we're going to have a look at a few to start with we're gonna have a look at a few in reverse and we're going to have a look at the differences in language as the sort of topic progresses just to make sure that you get your full marks and these sorts of questions but grab a piece of paper grab a pen and have a look at a couple of questions a few to have a go at but it's relatively quick and simple for us actually to do the maths behind this so it's not going to be that longer video so this one says there are 17 men and 26 women in a choir and the choir is going to sing at a concert it says one of the men and one of the women are going to be chosen to sing the first song work out the number of different pairs that can be chosen now if we kind of come away from this question for a second we think about maybe a bit of a simpler idea if we imagine instead maybe that there are three men and that there are two women and we have a look at the different amount of combinations that could happen here now for the sake of making this a bit more simple let's imagine that we've got man one man two and man three and we've got woman one and woman too and we kind of imagined this in a little bit of a table so man one at the top man two and man three and then we could have woman one over here and woman two here and we're gonna look at the different combinations that we could actually get if we think about the pairings that we could have so we could have a woman one paired with man one we could have one-on-one paired with man two and we could have the woman one paired with man three giving us three different combinations there then down below I've just realized that I've put an M there rather than a w so let's change that quickly I mean that was supposed to be woman two and then down below there we could have woman two also paired with man one we could have a woman too paired with Man Too and we could have a woman who paired with man three there we go so just a little bit of a visual idea for you but look if we count those up we've got one two three four five six and this sort of logic is going to be applied to all of these questions because really all we have to do if we knew we had three men and two women we just had to do three times two okay which would give us six options and this sort of logic is going to be applied to a lot of these questions when the numbers are bigger because obviously what we don't want to do is draw a column of 17 men going down and 26 women going across and have to draw out all of these combinations well actually distinct look every row is going to be every every amount of however many are in that row is just going to be multiplied by however many are in that column so we're going to apply that logic to a lot of these questions but there's just a basic understanding of why we're going to do this the way we do and how we can get away with applying a little trick here and not actually have to draw out all these combinations so coming back to this question look all we really care about is the fact there's 17 men and 26 women and we're going to do an amount of different combinations that can be chosen so it says work outly a number of different pairs that can't be chosen well nice and easy then all we have to do is think okay well there's 17 men that can be prepared with a potential 26 women so we're just going to do 17 times 26 there we go so 17 times 26 is 400 and 42 and there we go there's our final answer 442 different pairs that can be chosen so there we go that is the product rule for counting I'm gonna look at some different scenarios different wording and different sort of elements to this topic but essentially all it is is you take the number of options in the first pick and you multiply it by the amount of options in the second pick and in that second pick there there were 26 and in the first pick there were 17 so we just multiply those together let's have a look at a few different types of questions here okay so there is something slightly different in this one it says a cafe owner sells 10 different types of sandwiches Hassan buys a different type of Sandwich on Monday on Tuesday and on Wednesday and how many ways can you do this now this is slightly different look because it says he buys a different one A different type of Sandwich on each day but there's three days here so on the first pick if we imagine we've got our first pick here okay we've got 10 different types of sandwiches now by the time it gets to that second pick he no longer has 10 different types of sandwiches because it says he's going to buy a different type of Sandwich on each day so on that next day it doesn't matter what he buys on the first day but it'll only have nine options on the next day so we're gonna do ten times nine if it only said two days there our next would be 90 10 times nine so now we're thinking about obviously the third pick because it's three different days we've got Monday Tuesday and Wednesday and obviously if he's picked a sandwich on the first day he's picked a different one on the second day on that third day there he's not gonna have nine anymore but he's only gonna have eight to choose from just to make sure he gets a different sandwich on that third day as well so we've got 10 times 9 times 8 and that gives us a total I'm just doing these on the calculator of 720. there we go so we've got 720 different options okay now obviously uh the options here then the language is important it says how many different ways can you do this with a different type of Sandwich each day obviously let's imagine on the first day picks a cheese sandwich okay just think of a random sandwich here so if you pick the cheese sandwich on the first day um that would be one option obviously you can't have a cheese sandwich on the second or third day but if you picked a different type of Sandwich on the first day so not a cheese sandwich he could actually then get a cheese sandwich on the third day okay just not if he picks up on the first day so it might be that we've got the similar combinations here for example um you know the three same type of sandwiches but on different days so we don't have to worry about that it's just dependent on what he picks on what day in order for this sort of question here so we're gonna have a look at something slightly different as well but just thinking about this type of question here when it's when there's only one option of things to choose from this one being sandwiches rather than different people obviously we just need to make sure we've reduced that number by one if it is going to be a different one on the following day so just think about how many we actually could pick on that second and third day there let's have a look at having a go at some of these then so I've got a couple of questions for you to have a go out here they are okay so there's two questions here very similar to the first one that we first view that we've just looked at obviously just a different scenario here so have a little go at these two questions uh pause the video there and we'll go over the answer in a sec right okay so this first one says there are four starters seven Mains and four desserts in a restaurant work out the total number of ways of using a starter a main and a dessert so we've got three in here okay but it's not similar to that second one there's three but you know none of these are going to get reduced each time we've got four starters on the first pick on the second pick for the main we've got seven Mains and then we've got four desserts so it's just four times seven times four and that's going to give us 112 there we go so we've got 112 different ways of choosing from this particular menu there we go and there's our final answer now this next question here it says there are 52 cards in a deck okay James is going to give one to our list and then one to Ben how many different ways are there of doing this so on that first option there he's got 52 different cards that he can give to Alice by the time he's given one to Alice so one of those cards is gone and there's only 51 cards left there to give to Ben so it's going to be 51 options on the second pick so we just need to do 52 times 51. there we go making sure we reduce that by one and we get 2652 different ways that Ben can actually do that there we go and there's our final answer for that question right let's have a look at this in a little bit of a different way then okay so this question says Jeff is choosing a shrub and a rose tree for his garden at the Garden Center there are 17 different types of shrubs and some rose trees okay so we don't know how many rose trees Jeff says there are 215 different ways to choose one shrub and one rose tree could Jeff be correct and you must show how you get your answer now if we knew how many rose trees there were we would do 17 and we would multiply it by that number let's call that number X so 17 times x whatever that would be and that would give us the answer 215 if Jeff is correct here now basically all we need to figure out is is there a number that you can times 17 by to get 215 and if there is then obviously Jeff can be correct if there isn't then he can't be correct there it can't be possible so if we think about this in reverse then if we just do 215 and divide it by 17 I'm gonna do that on the calculator see what we get 215 divided by 17 and we get a decimal we get 12.64 and a lot more decimals there but 12.64 the important thing here is that it's not a whole number so if it was 12 for example 17 times 12 that give us 204 that could be that could be a possible amount 204 or maybe 17 times 13 which would give us 221 so they would both be possible combinations but this option here of having having 215 possible combinations is not correct so no okay because there is no number that multiplies by 17 to make 215. so just thinking about this in Reverse with the in terms of the total there and actually is that a possible total let's have a look at one more of these okay so in this question we've got some different options a lot more words it says Sadia is going to buy a new car for the car she can choose one body color one roof color and one wheel type and she can choose from 19 different body colors and 25 different wheel types but it doesn't then tell us how many which one does it not tell us it gives us the body color it gives us the wheel type but it doesn't tell us anything about the roof color so it says the total number of ways that she can choose these is 3325 work out the number of different roof colors that study can choose from so it doesn't say is it right or wrong there's definitely going to be 3325 we just need to figure out then what that number is going to be to actually get there so at the moment with these two we've got 19 and 25 so we know that just those we can do 19 times 25 and then we're going to times it by something else let's call it X again and that is going to equal 3325. so if we work out the 19 times 25 to start with that gives us 475 so let's just write this down here so we've got 475 find something is going to equal 3325. so we just need to figure out what that is and we can do it in reverse just like before we can do so 3325 divide that by 475 and it will tell us what that number is and this time we should get a hold number otherwise we've done something wrong so three three two five divided by 475 gives us the answer seven there we go so our answer is seven so we've got seven what is it about roof colors so we can say seven roof colors there we go that she can choose from and there's our final answer and just something something different here obviously thinking about it in reverse and making sure that we get that whole number there to figure out what number's missing in our multiplication right okay so quite nice and simple hopefully here's a couple for you to have a go up right okay so there's two questions here so have a go at these two pause the video there we'll go over the answers in a sec right okay so this first question says Tracy's going to choose a main course and a dessert in a restaurant she can choose from eight main courses and some desserts uh Tracy says there are 52 different ways to choose a Mainland a dessert could Tracy be correct well we've got eight uh main courses and we're going to multiply that by something and that's got to equal 52. so if we actually go about that in reverse then so 52 divided by 8 what does that equal hopefully you can already spot that without a calculator that's not going to be a whole number it comes out as 6.5 so no Tracy could not be correct because it would have to be either six or seven we can have 6.5 desserts okay so we could have just think about logically eight times six would give us 48 or we could have 8 times 7 which would give us 56. we definitely can't have 52 options there so there's our final answer for that first one and we'll just be no shown backed up by our working out there the next one Amelia is going to choose a new bike so you can choose from four different frame sizes 18 different colors and X different wheel types the total number of ways you can choose all three is 432 work out the number of different wheel types that Emilia can choose so we've got four different frames I'm going to multiply that by the 18 different colors and then we're going to multiply that by this unknown number that we don't have at the moment but it's going to equal 432. so 4 times 18 at the moment gives us a total of 72 so we've got 72 times something it's going to equal 432. so again just thinking about that in Reverse we'll do 432 divide that by 72 we'll see what we get 432 divided by that gives us the answer 6. there we go so there's six different wheel types there we go so six wheel types right there we go and there is our two answers for those two questions right let's have a look at something else and how this can sort of change with the language now okay so this question says there are 16 teams in a football league two teams are going to be chosen at random to play a match work out the number of different matches that could take place so when it comes to a question like this we've just got to be a little bit careful because it says here work out the number of different matches that could take place now obviously we've only got 16 teams now if we think about this obviously having something like three teams and let's just think about this in a slightly different way so if we had three teams and they're going to play some for these football matches let's just imagine we've got team one team two team three so again we've got team one team two team three there we are so obviously the actual teams can't play themselves so one against one is obviously not going to work so we'll just cancel out some of these at the moment two against two is not going to work and three against three doesn't work let's have a look at these options that we've got here so we could have one playing two we could have one playing three we're gonna have two playing one and two playing three or we could have three playing one or three playing two now obviously these are all the different matches that could take place if we were to pick them in any order but some of these matches are not different okay if we have a look we've got one against three which is the same as three against one so we can't have that that's not gonna be a different match that's going to be the same match so we can cancel off one of those then let's have a look at the next one let's go for a different color team one against team two is the same as team naught two against team one so obviously we're going to get rid of one of those let's get rid of that one and then we could also have and let's just see what other I've got I've got two against three and three against two and again one of those is going to be the same so we're going to need to get rid of one of those as well so let's get rid of that one so we've only got this one one against two we could have two against three or we could have three against one and if we apply the sort of logic here that we did before look let's think about what we do we've got three teams on the second pick we've only got two teams so we'd do three times two and that would give us the answer six but obviously half of those options were doubled up so they weren't different matches they were actually going to be the same match just in a slightly different order as they were happening at the same time okay we wouldn't we couldn't have the two two different options there so if we were to divide that well in in order to get us to get from six to two obviously it's half the amount so all we need to do is divide that by two and we get our answer there which is three okay so thinking about this obviously with the language there when it says the number of different matches that could take place obviously some of those are actually the same matches just in a slightly different order okay and we can't have that happening because that's not going to be a different match it's going to be exactly the same match it's just in terms of picking one of them first or picking one of them second but actually that does result in exactly the same match so actually if we have a look at this then in terms of our larger amount of teams here let's just get rid of all of this and again we don't want to draw out a big table for 16 teams because that's going to take absolutely ages but we could do that and obviously you'd see the same sort of logic being applied there but when we've got 16 teams look all we do is we have 16 teams on the first pick and then we are going to pick from 15 additional teams so we've got 16 times 15 to start with and 16 times 15 gives us a total result of 240 matches but obviously half of those matches are going to be exactly the same so they're not going to be different matches they're going to be the same match just picked in a different order so to finish this off we have to divide it by two 240 divided by 2 gives us 120 matches there we go so we've just got this little option here we just got to be careful sometimes with the language when it's just talking about something which can actually just be the same but in Reverse okay not like our sandwich uh scenario there where we can obviously have different options on different days but this is obviously happening at the same time so we can't have this same match um just in in the opposite of you know in Reverse it'd be exactly the same thing so obviously sometimes we just need to halve the answer here and we're gonna have a look at a couple of these sorts of questions okay so here's our next one Okay so we've got this question here it's very similar to one of the ones that we started on and it says there are 17 men in a choir the choir is going to sing at a concert two of the men are going to be chosen to bet um to make a pair to sing one of the songs Ben thinks the number of different pairs to be chosen is 136 Mark thinks the number of different pairs that can be chosen is 272 who is correct Ben or Mark give a reason for your answer so when we look at this look we've got the 17 men in a choir and it says two of them are going to be chosen okay so obviously if we're going to choose two of them and again you can apply the same logic as we did with the football matches when we actually picked these two people it doesn't matter whether I pick and it's called imagine it's man number one man number two up to 17. if I pick man number one and man number two that's gonna be exactly the same as picking man number two and man number one it doesn't matter what order they go in it's not happening on different days it's happening at the same time so I can't have two different options being the same two people okay so this is another option here where we're gonna have to halve our final result because on the first pick we've got 17 men and on the second pick we've got they're going to have 16 men to choose from and 17 times 16 gives us our answer 272 which is where Mark has got his answer there 272 but obviously we know that we can't have these options being doubled up because that would be exactly the same and it does say in the question here look different pairs different pairs and that's what we're looking for how many different pairs are there so we don't want one of the same pairs obviously being doubled up so we'll divide that by two and if we divide that by two we get the answer 136 there we go so we would say in this question Ben is correct okay and the reason being is that he hasn't included the doubled up pairs or something like that okay but there we go that is sort of what we would do here when we're looking for different pairs for something that's happening on the same day from the same sort of selection of people or something you know it could be selection of cards or something like that but here we go let's have a look at a couple of these questions have a go up right okay here we go says two questions so pause the video there have a go and we'll go over the answer in a sec right okay so this first one I left it says there are 52 cards in the deck Tom's going to give two cards to Jay how many different pairs of cards could Jay get so on the first pick he's got 52 and on the second pick he's got 51. there we go so 52 times 51 is going to give us this quite a large number here 2652 but obviously a lot of those options there if you know your cards well you could have the ace of hearts and the Ace of clubs let's say but that would be exactly the same as the Acer clubs and the Ace of Hearts okay just obviously in reverse and it does say how many different pairs does he get so we don't want no similar pairs there so we'll divide this by two there we go and we get the answer 1326 right there we go let's highlight that up and on our next one there are 30 students in a class two students are going to be selected to receive a prize how many different pairs there we go there's all the other languages same throughout them all how many different pairs of students could be selected so again on the first pick we've got 30. and then on the second pick we've got 29 so 30 times 29 gives us 870 different pairs but obviously again that is going to include you know people being doubled up on these two different selections so we also want to divide that by two there we go and we get 435 there we go and there is our final option right there we go so there's quite a few questions we've got two more questions for us to have a look at now something a little bit more challenging just in terms of the language and how we might approach it in different ways but we can have a look at some of these questions now right okay so this question says in a restaurant there are nine starters 15 Mains and eight desserts Janet is going to choose one of the following combinations for her meal so she's going to choose either a starter and a main a main and a starter I'm going to do this in difficult colors or a main and a dessert sorry or a final option is she's going to have a starter a mean and a dessert so all three show that there are a hundred and one thousand three hundred and thirty five different ways to choose the meal so for this first option then the starter and the main let's have a look at that to start with so for the starter and the main how can she do that so there's nine starters and 15 Mains that would be nine times fifteen there we go so 9 times 15 that's only different ways she has of doing that and that'll be 135 different options when it comes to the main and the dessert look let's have a look at that one second so there's 15 Mains and eight desserts that'd be 15 times eight and fifteen times eight there gives us 120. and then for this last one which I'll do over here we could have a starter a main and a dessert so that's going to be all three so that'd be nine times fifteen times the eight there we go so nine times fifteen times eight gives us one thousand and eighty there we go right so we've got all these different options let's just highlight these up we've got the 1080 the 135 and 120 and obviously it says sure that there are 1 335 different ways to choose this meal she's got all these different options to choose from so if we combine those all together we've got 1080 plus one three five plus the 120 and if we add those all up let's just double check it's definitely going to answer that there we go and that does equal one three three five there we go and that's shown there that there's 1335 different ways of choosing that meal if she's going to use one of those different combinations there right okay one more question to have a look at before we finish okay so something a little bit different here we've got two different questions it says there are three dials on a combination lock each dial can be set to one of the numbers one two three four or five the three digit number five five three which is on the diagram over here um is one way that it can be set so we can see that there five five three it says work out the number of different three digit numbers that can be set for the combination lock so it just says different three digit numbers so that can be any any amount of numbers on each lock let's have a look different three digit numbers and we could have one one one one one one two we can have any sort of different combination there it doesn't say anything about um not allowing allowing repeating numbers in each one so we could have one one one that'd be fine okay obviously not worrying about the fact that this five five three up there that's just one of the options that we could have within this but in terms of the different three digit numbers what could we have now Part B is slightly different it says how many how many of the possible three digit numbers have three different digits okay so three different digits there and that would mean that we couldn't have one one one anymore we could only have something with different digits so one two three would be fine but one one one would not be allowed because that's got the same digits within it so two different questions there so what I would say is pause the video have a go see what you get for these answers and we'll go over the answer in a sec okay so this first one it says work out the number of different three digit numbers so we just need to think about how many options we have on each pick so on the first pick we've got five different numbers to choose from so we've got five options there on the second pick it doesn't matter what we choose we can choose any of those five numbers again so we've got five numbers again and on the third pick again it doesn't matter what number we choose so we've just got five options again so it's five times five times five and that would give us a hundred and twenty five different options there different combinations that we can set on this padlock and there we go there's our first answer now the second one's slightly different it says how many of the possible three digit numbers have three different digits so this means on the first pick look we've got five options but on the next pick Let's imagine we've picked a one we cannot have a one again so we can have a two three four or five so we've only got four options to pick from now so that would be times four and then let's imagine we've picked a one and a two we can no longer have a one and a two so we can only pick a three four or five so we've only got three options now on that third pick so that'd be three options so five times four times three gives us a total of sixty options if we want to make sure we have all different digits there all right there we go and that is our final answer on this last question so that is just an overview of all the different kinds of questions that you could have on the product rule for counting so hopefully you found that useful if you did if it was helpful please like please comment Please Subscribe and I'll see you for the next one [Music] [Music]