### Video Transcript

In part one, we looked at how to count up the number of ways to arrange objects and use the factorial notation to produce a formula for this. We also looked at counting how many ways there are to pick a certain number of objects from a set and came up with the permutations formula, ๐ ๐ ๐. Now weโre gonna use these things to count combinations of objects where their order isnโt relevant and weโre gonna use the combinations formula, ๐ ๐ถ ๐ in a few examples.

So remember, we learned last time that ๐ factorial means ๐ times ๐ minus one times ๐ minus two, and so on and so on, times three, times two, all the way down to times one. So, for example, five factorial means five times four times three times two times one. And we also learned the formula ๐ ๐ ๐, permutations formula, is equal to ๐ factorial over ๐ minus ๐ factorial, so weโre picking ๐ objects from a set of ๐ objects. Now we did say that zero factorial is equal to one. So if ๐ and ๐ turn out to be equal, it would mean that with zero factorial on the bottom, weโd just make that equal to one. Well an example of that is in the question how many ways are there to pick three letters from the letters ๐ด, ๐ต, ๐ถ, ๐ท.

So in this case ๐ would be equal to three cause thatโs the number of letters weโre choosing, and ๐ would be equal to four cause thatโs the size of the set of letters that weโre choosing from. So ๐ would be three because that is the number of letters weโre choosing, and ๐ would be equal to four because thatโs the size of the set of letters that weโre choosing from. So the answer to the question would be four ๐ three, so thatโs four factorial over four minus three factorial, which of course is four factorial over one factorial. And when we worked that out, we get an answer of twenty-four. So the important thing about this is that the order of the letters was quite important because that twenty-four combinations there includes all of these ones here. And these are basically all variations on ๐ด, ๐ต, and ๐ถ, so just add the letters ๐ด, ๐ต, and ๐ถ in different orders. If it doesnโt matter to you what order theyโre in, effectively we can count all of those different combinations as one choice, just one option. And itโs that process that weโre going to look at in this video.

So, as weโve just seen, ๐ ๐ ๐ tells us how many permutations there are for picking ๐ objects from a set of ๐ objects. And again, as weโve just seen, each group of ๐ letters โ so we have three letters in that example ๐ด, ๐ต, and, ๐ถ โ is written ๐ factorial different ways. So three factorial different ways. Thatโs three times two times one is six different ways in the big list. So that big list contains a lot of repeats if youโre not really interested in what order they appear in. So weโre gonna look at how to work out how many different combinations we get when choosing ๐ objects from set of ๐ objects if we discard all the different rearrangements of the same letter. So ๐ ๐ ๐, as weโve said, told us how many unique permutations we had, and that list included ๐ factorial repeats of groups of three letters. So if we just want to know for example how many groups of three letters, or ๐ letters, that weโve got in a final answer, then what we need to do is take the answer ๐ ๐ ๐ and divide it by ๐ factorial. So there we are, ๐ ๐ถ ๐ is ๐ ๐ ๐ divided by ๐ factorial. So the formula for ๐ ๐ ๐, remember, was ๐ factorial over ๐ minus ๐ factorial. If we divide that by ๐ factorial, then itโs the same as multiplying by one over ๐ factorial and that gives us this formula here, ๐ factorial over ๐ factorial ๐ minus ๐ factorial. So remember, ๐ ๐ ๐ counts the permutations where different orders of the same group of letters count separately, so weโve kind of kept double, triple, quadruple counting things and so on. And ๐ ๐ถ ๐ counts the combinations where different orderings of the group are combined together. So in our last example, all the ๐ด๐ต๐ถ, ๐ด๐ถ๐ต, ๐ต๐ด๐ถ, and so on, that would count as multiple permutations in the ๐ ๐ ๐ formula, but we would count all six of those variations just as one combination in the ๐ ๐ถ ๐ formula because they are all the same group of three letters.

Right, letโs have a look at a couple of examples then. So if weโve got ten discs labelled ๐ด up to ๐ฝ, we take three discs at random and set them down in the order that they were selected. How many permutations are there for this? So ๐ would be ten in this case because weโve got a set of ten letters to choose from. And ๐ will be three because weโre picking three from that set. Now weโre setting them down in order, so ๐ด๐ต๐ถ for example would not be equivalent to ๐ถ๐ต๐ด and so on, so we want to use the ๐ ๐ ๐ formula.

And substituting in those values for ๐ and ๐, weโve got ten ๐ three, so thatโs ten factorial over ten minus three factorial. And that gives us ten factorial over seven factorial. Now, obviously, we could just put that in our calculator and get our answer, but Iโm just gonna write that out in full for a moment. And when we do that, we can see that we can do quite a lot of cancelling. So seven divided by seven is one, six divided by six is one, five divided by five is one, four divided by four is one, three divided by three is one, and so on. All of those cancel out, so weโre ending up with just ten times nine times eight over one or just ten times nine times eight. So that gives us seven hundred and twenty different permutations that we can get. So thatโs our answer. Now, just to sort of highlight the difference between ๐ ๐ ๐ and ๐ ๐ถ ๐, remember that seven hundred and twenty includes for every set of three letters like ๐ด๐ต๐ถ; weโve got six variations on that in there. And if weโre prepared to say that theyโre all equivalent cause theyโre just combinations of the letters ๐ด, ๐ต, and ๐ถ, we can divide that seven hundred and twenty by six, in this case, three factorial because we had ๐ was equal to three to work out the number of combinations. So thereโs the calculation then for ๐ ๐ถ ๐. If we were looking for how many clusters of three letters, the answer would only be a hundred and twenty. But the question specifically asked for permutations and it said that the order was important, so we settled down in the order that they were selected. So this is the correct answer.

Just make sure that, on the page, weโve got the-the proper correct answer highlighted, and itโs obvious that this other working out wasnโt just another guess of what the answer might be. Okay, letโs look at this small essay for our next question. In a lottery, we choose five letters of the alphabet. During the draw a machine loads twenty-six balls, each way- each one has one of the letters ๐ด to ๐ on them. It shuffles them and then it lets out five. It doesnโt matter what order they were drawn in if our five chosen letters match the machines, then we win! How many combinations of five letters are there to choose from? So in this case the order doesnโt matter, so weโre gonna be using the ๐ ๐ถ ๐ combination. And there are twenty-six letters of the alphabet to choose from, so ๐ is equal to twenty-six. And weโre choosing five of those letters, so ๐ is equal to five. So weโre choosing five, ๐ equals five, from twenty-six, ๐ equals twenty-six. And it doesnโt matter what order they were drawn in, so weโre using the ๐ ๐ถ ๐ formula which is twenty-six ๐ถ five when we plug those values in for ๐ and ๐. So that gives us twenty-six factorial over five factorial twenty-six minus five factorial, which is twenty-six factorial over five factorial twenty-one factorial. Now again we can just plug this number into a calculator and get our answer, but Iโm just gonna take a little extra step here and just kind of split this out slightly. Sometimes the numbers that you get are too big to work on a calculator; but by spotting some things that you can cancel, you can actually generate numbers which are small enough that your calculator can still handle. And thatโs, well, thatโs not the case here. We can, you know, this-this number would work on a calculator, but letโs see the technique anyway. So havenโt quite got room to write out twenty-six factorial in full but I know itโs twenty-six times twenty-five times twenty-four times twenty-three times twenty-two times twenty-one, and so on and so on. But of course this bit here, twenty-one times all the numbers down to one, is just twenty-one factorial. So weโve got this stuff here, times twenty-one factorial on there-on the top and on the denominator. Weโve got a twenty-one factorial, so in fact these twenty-one factorials will cancel out. So weโve got this divided by five factorial. So sixty-five thousand seven hundred and eighty, thatโs the number of combinations that are there. So if this whole thing was done at random, thatโs the number of different ways there are of choosing five letters, so weโve got one in sixty-five thousand seven hundred and eighty chance of winning this particular lottery. Now as a bit of an aside, if it did matter, if we had to match the order in which the letters came out as well, we wouldโve used the ๐ ๐ ๐ formula. And of course weโve got groups of five, so thatโs gonna five factorial sort of different combinations of each grouping of five letters, so the answer is gonna be five factorial times bigger than that: seven million eight hundred ninety-three thousand six hundred. So actually youโve got a far smaller chance of winning that particular lottery if it does matter what order the balls came out in, but of course thatโs not the answer we were looking for in this particular case.

Okay, finally then, letโs just use our combination counting techniques to calculate some probabilities. So weโve got a question here. A bag contains five chocolate and fifteen strawberry sweets. If someone chooses a sweet at random, find the probability that they pick a chocolate one; and if two people choose a sweet, each at random, find the probability that they both get chocolate. So really weโve got two questions, letโs call them ๐ด and ๐ต. Letโs look at the first one first. Someone chooses a sweet at random, find the probability itโs chocolate. Well, this is pretty straightforward. The probability you get a chocolate sweet is just the number of ways of getting a chocolate sweet divided by the number of ways of choosing a sweet of any sort. And of course there are five chocolate sweets and then there are fiv- fifteen strawberry sweets, so itโs five divided by five plus fifteen, so thatโs five over twenty. So our answer, the probability of getting a chocolate sweet, is just five over twenty. Okay, letโs look at the second question. Now weโre going to look at two different methods of doing this question. The first method, which is maybe the one that youโve already thought of, weโve got two events. So first of all person one gets a chocolate sweet and then person two gets a chocolate sweet. So for those two things to happen together, we have to have one and the other, weโre going to multiply those probabilities together. So weโve just worked out that the probability of one person getting a chocolate sweet is five over twenty. And of course if theyโve picked a chocolate sweet, one, thereโs just gonna be four chocolate sweets left, and the other thing, thereโll now only be nineteen sweets in total for them to choose from. So weโre gonna end up with five-twentieths as the probability of the first person getting chocolate and four-ninetieths as a probability of the second person getting a chocolate. Multiply those two together, do a bit of cancelling, and we get one over nineteen. Again, in probability, youโre not gonna get penalised if you donโt cancel those numbers down. But in this case, this three hundred and eighty, has may be lost a little bit of its meaning, so itโs probably more sensible to cancel down to those nice and simple numbers, one over nineteen. So method two then, the probability they both get chocolate is the number of ways that two people can choose chocolate divided by the number of ways that two people can just choose any sweet. So the number of ways that two people can choose chocolate is five ๐ถ two which would be choosing from five chocolates, and thereโs two of them, so five ๐ถ two is that. And the number of ways that they can choose a sweet, but weโve got twenty sweets in total and thereโs two of them choosing, so the total number of combinations of ways they can choose is twenty ๐ถ two. And if you do this on your calculator, five ๐ถ two turns out to be ten and twenty ๐ถ two turns out to be a hundred and ninety, which reassuringly when we cancel down we get the same answer as we did the other way, one over nineteen. So thereโs a one in nineteen chance that both of them will get chocolate.

So just to summarise what weโve learned over the course of these two videos, weโve learned about ๐ factorial, which has got different ways of expressing it, but itโs ๐ times the number one smaller times the number one smaller, and so on and so on, until we get down to one. And remember, weโve got that special definition of zero factorial is actually equal to one; so not entirely intuitive, but you have to remember that. Weโve learned about permutations, so ๐ ๐ ๐ and the various different ways of notating that in different regions, ๐ factorial over ๐ minus ๐ factorial. Now this counts every different rearrangement of the same groupings as letters as individual cases, so we get quite a large number in that case. But if we want to merge together all different arrangement of the same letters that weโve chosen and get a smaller answer, we have to divide that by ๐ factorial. And because weโve got all those repeats of the same combinations of those letters or objects, we get combinations; we get the ๐ ๐ถ ๐ formula. And thereโs different ways again of notating that, and that gives us the formula ๐ factorial over ๐ factorial ๐ minus ๐ factorial.