1 00:00:00,120 --> 00:00:05,130 So we're going to start off here by talking about factory goals because they are foundational to the 2 00:00:05,130 --> 00:00:12,300 concepts of permutations and combinations which we'll need to get into later in this combinatorics section. 3 00:00:12,420 --> 00:00:20,400 The idea of a factorial is that when we take any positive integer or the number zero and we apply the 4 00:00:20,400 --> 00:00:21,660 factorial operator. 5 00:00:21,660 --> 00:00:26,780 So let's say, for instance here that we're taking three factorial, it's this three exclamation point 6 00:00:26,790 --> 00:00:33,750 Applying this factorial operator to the number three means take three and multiply it by all of the 7 00:00:33,750 --> 00:00:37,920 positive integers, less than three, all the way down to one. 8 00:00:37,920 --> 00:00:43,680 So three factorial is equivalent to three times two times one. 9 00:00:43,680 --> 00:00:49,110 And it doesn't matter what number we're starting with here, we always continue to expand the product 10 00:00:49,110 --> 00:00:51,050 until we get down to one. 11 00:00:51,060 --> 00:00:57,120 So for instance, seven factorial is the same product, except we're starting with seven and multiplying 12 00:00:57,120 --> 00:01:02,100 all the way down to one in the same way that for three factorial we started at three and multiplied 13 00:01:02,100 --> 00:01:03,390 all the way down to one. 14 00:01:03,390 --> 00:01:06,390 And the same thing applies for much larger numbers. 15 00:01:06,390 --> 00:01:09,870 100 factorial will follow the same pattern. 16 00:01:09,870 --> 00:01:16,200 We'll start with 100 and multiply by 99, 98, 97, 96 and we'll continue on all the way down until 17 00:01:16,200 --> 00:01:19,290 we get to three times, two times one where we stop. 18 00:01:19,290 --> 00:01:22,110 So that's the concept of factorial. 19 00:01:22,110 --> 00:01:27,540 And it's important to note that we have factorial nested within other factorial. 20 00:01:27,540 --> 00:01:33,570 For instance, if we look here at seven factorial and we just consider everything in the product other 21 00:01:33,570 --> 00:01:34,500 than the seven. 22 00:01:34,500 --> 00:01:37,860 So we look at six times, five times four, all the way down to one. 23 00:01:37,860 --> 00:01:45,570 We notice that this part is six factorial, which tells us that seven factorial is equal to seven times 24 00:01:45,570 --> 00:01:46,710 six factorial. 25 00:01:46,710 --> 00:01:50,400 And that brings us to this property of factorial. 26 00:01:50,400 --> 00:01:54,780 So what we see here is that we can plug any positive integer in for N. 27 00:01:54,780 --> 00:01:57,840 So for instance, let's say we want to use n equals. 28 00:01:58,490 --> 00:01:59,320 20. 29 00:01:59,330 --> 00:02:06,800 If we plug this value into this property, we get 20 factorial is equal to and is 20. 30 00:02:06,800 --> 00:02:10,850 So we get 20 multiplied by 20, minus one is 19. 31 00:02:10,850 --> 00:02:15,160 So we're multiplying here by 19 factorial. 32 00:02:15,170 --> 00:02:22,100 So 20 factorial is equal to 20 times 19 factorial in the same way that we saw here that seven factorial 33 00:02:22,100 --> 00:02:29,600 is equal to seven times six factorial or that 100 factorial is equal to 100 times 99 factorial. 34 00:02:29,810 --> 00:02:36,500 And of course, we can use this property to continue expanding this factorial expression because this 35 00:02:36,500 --> 00:02:40,130 property tells us that 19 factorial is the same. 36 00:02:41,110 --> 00:02:43,720 As 19 times 18 factorial. 37 00:02:43,720 --> 00:02:51,160 So we could also rewrite this as 19 times 18 factorial, and we could continue to apply that property 38 00:02:51,160 --> 00:02:57,370 to rewrite 18 factorial and then 17 factorial until eventually we ended up at the full expansion of 39 00:02:57,370 --> 00:02:58,690 20 factorial. 40 00:02:58,690 --> 00:03:00,220 So that's the idea here. 41 00:03:00,220 --> 00:03:02,980 But what are factorial is really good for? 42 00:03:02,980 --> 00:03:04,720 What are they actually telling us? 43 00:03:04,720 --> 00:03:11,050 Well, we can think about the real world concept of a factorial as the number of ways in which we can 44 00:03:11,050 --> 00:03:17,080 arrange some set of objects or people or data, whatever kind of set that we're working with. 45 00:03:17,080 --> 00:03:23,320 For instance, let's think about playing cards in particular, a standard deck of 52 cards. 46 00:03:23,320 --> 00:03:26,860 We know it has face cards, kings, queens and jacks. 47 00:03:26,860 --> 00:03:29,110 Let's say that we just have one. 48 00:03:29,110 --> 00:03:33,400 KING In other words, of the 52 cards, we only have one card. 49 00:03:33,400 --> 00:03:34,240 It's a face card. 50 00:03:34,240 --> 00:03:34,930 It's a king. 51 00:03:34,930 --> 00:03:41,470 If we lay this king on the table, what we realize is that there's only one arrangement we can make 52 00:03:41,470 --> 00:03:45,580 with this KING If we lay it on the table, it's just the king by itself. 53 00:03:45,580 --> 00:03:49,090 There's only one way in which we can arrange that single card. 54 00:03:49,390 --> 00:03:52,420 But now let's say that we have a set of two cards. 55 00:03:52,420 --> 00:03:58,030 We have a king and a queen, and we want to find out the number of ways we can arrange this on the table. 56 00:03:58,030 --> 00:04:03,730 Well, we can either lay down the king first and then the queen, or alternatively, we have a second 57 00:04:03,730 --> 00:04:04,300 arrangement. 58 00:04:04,300 --> 00:04:09,160 We could lay down the queen first and then we could lay down the king. 59 00:04:09,430 --> 00:04:13,990 And so when we have two cards, there are two ways in which we could arrange them. 60 00:04:13,990 --> 00:04:18,370 What happens when we have three cards, one king, one queen and one Jack? 61 00:04:18,370 --> 00:04:23,440 Well, we realize that we could lay down the king first, and then after we lay down the king, we have 62 00:04:23,440 --> 00:04:28,270 two options either the queen, then the jack or the jack, then the queen. 63 00:04:28,270 --> 00:04:36,820 So we could keep the king in the first position and then lay down the jack instead, followed by the 64 00:04:36,820 --> 00:04:37,390 queen. 65 00:04:37,750 --> 00:04:41,470 But these are the only two options we have if we lay the king down first. 66 00:04:41,470 --> 00:04:44,170 Once the king is gone, we just have two cards left. 67 00:04:44,170 --> 00:04:46,960 There's only two ways to arrange those two cards. 68 00:04:46,960 --> 00:04:50,350 First, the queen than the Jack or first the jack than the Queen. 69 00:04:50,350 --> 00:04:52,180 So our only options are king. 70 00:04:52,180 --> 00:04:54,190 Queen Jack or King Jack. 71 00:04:54,190 --> 00:04:57,370 Queen when the king comes first. 72 00:04:57,370 --> 00:05:02,410 But of course, we could also put the queen first or the jack first. 73 00:05:02,410 --> 00:05:05,800 So let's look at that if we put the queen first. 74 00:05:06,600 --> 00:05:10,440 Then as you can start to see, we have two options. 75 00:05:10,440 --> 00:05:18,490 We can either put the king second or the king third with the Jack third or the jack second. 76 00:05:18,510 --> 00:05:19,850 So two options here. 77 00:05:19,860 --> 00:05:22,290 Queen King Jack or Queen Jack King. 78 00:05:22,440 --> 00:05:26,890 And then our last pair of options is Jack first. 79 00:05:26,910 --> 00:05:32,190 We could put the jack first and then we could follow it up with the king, then the queen. 80 00:05:33,030 --> 00:05:36,910 Or the queen and then the king. 81 00:05:36,930 --> 00:05:43,170 And these, in fact, are all of the options we have for arranging three cards down on the table. 82 00:05:43,200 --> 00:05:49,230 What we want to realize here is that when we have three cards, we have six possibilities. 83 00:05:49,230 --> 00:05:52,140 When we have two cards, we have two possibilities. 84 00:05:52,140 --> 00:05:54,870 And when we have one card, we have one possibility. 85 00:05:55,200 --> 00:06:02,160 What we see, though, is that six is actually equivalent to we could rewrite it as three times, two 86 00:06:02,160 --> 00:06:03,000 times one. 87 00:06:03,330 --> 00:06:10,290 We could also rewrite two as two times one, and we could rewrite one as just one or keep it the same, 88 00:06:10,290 --> 00:06:11,070 I should say. 89 00:06:11,280 --> 00:06:16,290 So what we see then is that this is equivalent to three factorial. 90 00:06:16,530 --> 00:06:19,260 This is equivalent to two factorial. 91 00:06:19,350 --> 00:06:22,920 And this then, of course, is equivalent to one factorial. 92 00:06:22,920 --> 00:06:28,860 And that's why we say that this factorial operation tells us the number of ways in which we can arrange 93 00:06:28,860 --> 00:06:32,700 the set of objects, people, data, etc. that we're working with. 94 00:06:32,910 --> 00:06:38,280 So we have this idea that the number of ways in which we can arrange three cards is three factorial, 95 00:06:38,310 --> 00:06:42,810 that the number of ways in which we can arrange two cards is two factorial, and that if we have one 96 00:06:42,810 --> 00:06:46,200 card, the number of arrangements is given by one factorial. 97 00:06:46,440 --> 00:06:48,730 So let's take this one step further. 98 00:06:48,750 --> 00:06:50,880 What happens when we have zero cards? 99 00:06:50,880 --> 00:06:52,560 We have no cards at all. 100 00:06:52,710 --> 00:06:55,800 In how many ways can we arrange zero cards? 101 00:06:55,830 --> 00:07:00,060 Well, if you think about it, if we're laying these cards on the table and we have no cards at all, 102 00:07:00,090 --> 00:07:02,850 there's only one possible arrangement here. 103 00:07:02,850 --> 00:07:11,280 And that one possible arrangement is no cards at all on the table, which suggests to us that zero factorial 104 00:07:12,060 --> 00:07:13,740 is actually equal to one. 105 00:07:13,740 --> 00:07:14,730 And that's true. 106 00:07:14,730 --> 00:07:19,560 Even though it may seem counterintuitive at first, we define zero factorial as equal to one. 107 00:07:19,560 --> 00:07:26,370 We can see that using this kind of a real world pattern, we can also see it by looking at this pattern 108 00:07:26,370 --> 00:07:30,870 we've lined up here, these factorial values, and we've already shown that some of these others here 109 00:07:30,870 --> 00:07:31,800 are true. 110 00:07:31,980 --> 00:07:36,960 So if we call this value on the left and factorial, let's say this is n factorial. 111 00:07:36,960 --> 00:07:39,480 So we have an equal to four three, 210. 112 00:07:39,480 --> 00:07:46,830 What we notice here is that if we start at 24 and we divide it by four, we get to six, 24 divided 113 00:07:46,830 --> 00:07:47,670 by four is six. 114 00:07:47,670 --> 00:07:49,170 And that's this value right here. 115 00:07:49,170 --> 00:07:53,160 So we get 24 divided by four is equal to six. 116 00:07:53,160 --> 00:07:59,970 We take six divided by three and we get to two, we take two divided by two and we get to one. 117 00:08:00,060 --> 00:08:07,080 And we see this pattern emerging 24 divided by 4 to 6, six divided by 3 to 2, two divided by 2 to 118 00:08:07,080 --> 00:08:14,400 1, which means that we should be able to say one divided by one gives us whichever value should go 119 00:08:14,640 --> 00:08:18,360 here, one divided by one gives us this value. 120 00:08:18,360 --> 00:08:22,500 And so we have to say then that this value must be equal to one. 121 00:08:22,500 --> 00:08:25,800 And that's another way to show that zero factorial is equal to one. 122 00:08:26,100 --> 00:08:31,500 What happens if we follow this pattern and we try to define the factorial operation for a negative integer 123 00:08:31,500 --> 00:08:32,610 like negative one? 124 00:08:32,610 --> 00:08:41,010 Well, if we say here we have negative one factorial, then to get the value of negative one factorial 125 00:08:41,010 --> 00:08:47,100 right here, following our same pattern, we would say one divided by zero should give us this value 126 00:08:47,100 --> 00:08:47,790 right here. 127 00:08:48,060 --> 00:08:51,390 But of course in mathematics we can't divide by zero. 128 00:08:51,390 --> 00:08:59,400 Dividing by zero is undefined, which means that this value here has to be undefined. 129 00:08:59,400 --> 00:09:05,550 And this in part is why we say that the factorial operator is not defined for negative integers, it's 130 00:09:05,550 --> 00:09:10,170 only defined or we only define it for zero and positive integers. 131 00:09:10,170 --> 00:09:15,480 So with all that being said, now that we have a better idea of what a factorial is and how to use it, 132 00:09:15,480 --> 00:09:18,630 let's talk briefly about some factorial operations. 133 00:09:19,140 --> 00:09:24,090 Specifically, let's look at addition, subtraction, multiplication and division of factorial. 134 00:09:24,090 --> 00:09:30,690 And we'll start with division because division will be the most common operation that we'll use when 135 00:09:30,690 --> 00:09:33,090 we talk about permutations and combinations. 136 00:09:33,090 --> 00:09:39,210 So for instance, if we have four factorial divided by two factorial, remember, if you ever get stuck 137 00:09:39,210 --> 00:09:43,830 simplifying factorial, you can always write them out or at least parts of them. 138 00:09:43,830 --> 00:09:49,560 It'll give you a better idea of how we can simplify something like this factorial division problem. 139 00:09:49,740 --> 00:09:55,530 So if we write this out, we can say that it's equal to four times, three times, two times one, and 140 00:09:55,530 --> 00:09:58,200 we're dividing that by two times one. 141 00:09:58,200 --> 00:10:03,600 Well, whenever we have the same common factor in the numerator and the denominator of a fraction, 142 00:10:03,600 --> 00:10:08,940 we can cancel it, which means that we can cancel these twos with one another and we can cancel these 143 00:10:08,940 --> 00:10:10,170 ones with one another. 144 00:10:10,170 --> 00:10:14,730 And so all we're left with is 12 in the numerator and one in the denominator. 145 00:10:14,730 --> 00:10:17,580 12 divided by one is 12. 146 00:10:17,880 --> 00:10:22,380 And of course, the same thing would go for the reciprocal of this fraction. 147 00:10:22,410 --> 00:10:28,590 If we flip it upside down and we take two factorial divided by four factorial, we get two times one 148 00:10:28,590 --> 00:10:32,430 and in the denominator four times three times. 149 00:10:32,460 --> 00:10:33,660 Times two times one. 150 00:10:33,780 --> 00:10:40,520 And so, again, we can cancel the twos and the ones and all we're left with. 151 00:10:40,530 --> 00:10:45,960 Then whenever you cancel everything from one side of a fraction, all you're left with is an understood 152 00:10:45,960 --> 00:10:46,750 one. 153 00:10:46,770 --> 00:10:49,680 So we say that this is one divided by 12. 154 00:10:49,710 --> 00:10:55,410 In other words, whenever you're dividing two factorial as one factorial by another, we realize that 155 00:10:55,410 --> 00:10:59,310 the smaller factorial is going to cancel completely. 156 00:10:59,310 --> 00:11:04,770 Two is the smaller factorial in both of these cases and notice how its entire product cancelled completely 157 00:11:04,770 --> 00:11:10,680 this two times one and this two times one went away because of course it's sort of embedded in this 158 00:11:10,680 --> 00:11:11,760 four factorial. 159 00:11:11,760 --> 00:11:16,920 We're also going to have the same two times one as part of the four factorial, so that smaller factorial 160 00:11:16,920 --> 00:11:18,510 will always cancel completely. 161 00:11:18,510 --> 00:11:24,780 All we'll be left with is the part of the product of the larger factorial that extends all the way down 162 00:11:24,780 --> 00:11:25,980 to the smaller one. 163 00:11:25,980 --> 00:11:32,550 So we start here at four and we say four times three and then we stop because we've arrived at two, 164 00:11:32,550 --> 00:11:35,460 which is the starting point of this smaller factorial. 165 00:11:35,670 --> 00:11:37,020 So that's division. 166 00:11:37,020 --> 00:11:39,660 Multiplication is just as simple. 167 00:11:39,660 --> 00:11:46,020 If we take four factorial and we multiply it by two factorial, usually the easiest way to go about 168 00:11:46,020 --> 00:11:52,680 simplifying here is just to say that four factorial is 24 and two factorial is two. 169 00:11:52,800 --> 00:11:56,160 So we get 24 times two or 48. 170 00:11:56,460 --> 00:11:59,790 Now that being said, we could also do some algebra here. 171 00:11:59,790 --> 00:12:06,300 We could write these out four times, three times, two times, one multiplied by two times one. 172 00:12:06,300 --> 00:12:13,020 We could then acknowledge here that the two times one factor occurs twice. 173 00:12:13,020 --> 00:12:15,000 Everything here is multiplied together. 174 00:12:15,060 --> 00:12:21,210 So we could rewrite this two times, one times, two times one as two times one squared instead. 175 00:12:21,210 --> 00:12:23,160 So rewriting it this way. 176 00:12:24,300 --> 00:12:27,120 Is the same thing two times one quantity squared. 177 00:12:27,240 --> 00:12:33,390 So we would then get 12 multiplied by two squared or. 178 00:12:34,070 --> 00:12:37,730 12 times four or 48. 179 00:12:37,760 --> 00:12:41,540 That's obviously a slower way to get to this answer of 48. 180 00:12:41,540 --> 00:12:45,830 But it shows you the math in different ways so we can understand what's actually happening here. 181 00:12:45,860 --> 00:12:54,710 Now, if we look at, for instance, subtraction of factorial, if we say four factorial minus two factorial, 182 00:12:54,740 --> 00:13:00,680 again, we can do it the simple way here where this is equal to 24 minus two. 183 00:13:01,340 --> 00:13:02,540 Or 22. 184 00:13:02,660 --> 00:13:05,600 Or we could again, use some factoring. 185 00:13:05,600 --> 00:13:12,500 We could say four times, three times, two times one minus two times one. 186 00:13:12,500 --> 00:13:15,710 And we see that we have this two times one common factor. 187 00:13:15,800 --> 00:13:18,920 So we could pull that out in front, two times one. 188 00:13:19,190 --> 00:13:22,790 All we're left with from this first term is four times three. 189 00:13:23,460 --> 00:13:26,490 And all we're left with from the second term is an understood one. 190 00:13:26,490 --> 00:13:28,620 So we get minus one here. 191 00:13:28,860 --> 00:13:36,600 And that just leaves us with two times 12 minus one or two times 11 or 22. 192 00:13:36,600 --> 00:13:39,030 And we see that we get back to this same value. 193 00:13:39,450 --> 00:13:41,580 What happens, though, if we do addition? 194 00:13:41,580 --> 00:13:44,820 Well, the operation is really going to look exactly the same. 195 00:13:44,820 --> 00:13:50,580 If we're adding instead, we would get 24 plus two or 26. 196 00:13:50,610 --> 00:13:51,660 And same thing here. 197 00:13:51,660 --> 00:13:54,810 We would then add we would factor out the two times one. 198 00:13:54,810 --> 00:14:00,630 We would be left with four times three plus one or two times 12 plus one. 199 00:14:00,630 --> 00:14:06,480 And that would leave us instead with two times 13 or 26. 200 00:14:06,660 --> 00:14:09,480 The same answer we got when we simplified it the easy way. 201 00:14:09,480 --> 00:14:13,960 So hopefully that gives you a better feel for how we work with factorial. 202 00:14:14,040 --> 00:14:17,700 By looking at some of these arithmetic operations we can do with them. 203 00:14:18,030 --> 00:14:23,910 Keep in mind that we've been working with a lot of small numbers here, but factorial can get big really 204 00:14:23,910 --> 00:14:24,390 quickly. 205 00:14:24,390 --> 00:14:30,690 We know that zero factorial is one, one factorial is one, two factorial is two, three, factorial 206 00:14:30,690 --> 00:14:31,440 is six. 207 00:14:31,440 --> 00:14:40,830 And if we keep going, we get 24, 120 720 and we can see how these factorial values start to get big 208 00:14:40,830 --> 00:14:41,910 really quickly. 209 00:14:41,910 --> 00:14:48,750 So because of that, we just assume that we will of course be using a calculator or a computer to calculate 210 00:14:48,750 --> 00:14:50,790 some of these larger factorial cells. 211 00:14:50,790 --> 00:14:55,740 Really, any scientific calculator will have a factorial button or some kind of factorial operation. 212 00:14:55,740 --> 00:14:58,800 Or you can always just Google factorial calculator. 213 00:14:58,800 --> 00:15:05,970 But thinking about how large these numbers really are, if you look any deeper into factorial math, 214 00:15:05,970 --> 00:15:14,070 52 factorial is a common example that you'll run into because we have a standard 52 card deck like we 215 00:15:14,070 --> 00:15:16,620 talked about earlier with the Kings, queens and Jacks. 216 00:15:16,620 --> 00:15:22,740 But 52 factorial is the number of ways in which you can arrange a standard 52 card deck of cards. 217 00:15:22,740 --> 00:15:30,000 But this is a number that is so incomprehensibly large that if you give a deck of cards a perfect shuffle, 218 00:15:30,030 --> 00:15:35,700 the arrangement you create with that shuffle is virtually guaranteed to be a unique shuffle that has 219 00:15:35,700 --> 00:15:38,670 never been seen before in all of human history. 220 00:15:38,670 --> 00:15:43,170 You'll also see the 70 factorial example. 221 00:15:43,200 --> 00:15:48,570 70 factorial is a number that is just larger than what we call a Google. 222 00:15:48,600 --> 00:15:52,050 You're already familiar with million, billion, trillion, etc.. 223 00:15:52,050 --> 00:15:56,250 Well, a Google is the number one followed by. 224 00:15:56,250 --> 00:15:59,970 So after this one followed by 100. 225 00:16:00,770 --> 00:16:01,640 Zeroes. 226 00:16:01,820 --> 00:16:08,420 So if you think about the number one and then 100 zeroes after it, that number is a Google and 70 factorial 227 00:16:08,420 --> 00:16:11,450 is just larger than one Google. 228 00:16:11,480 --> 00:16:18,020 And as you may have already guessed, that Google number, that extremely large, that incomprehensibly 229 00:16:18,020 --> 00:16:24,200 large number one, Google is, of course, where the company Google got its name.