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All language subtitles for 010 A Recap of Combinatorics_en

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Would you like to inspect the original subtitles? These are the user uploaded subtitles that are being translated: 1 00:00:03,030 --> 00:00:04,290 Narrator: Hello, everyone. 2 00:00:04,290 --> 00:00:05,790 The purpose of this video is 3 00:00:05,790 --> 00:00:08,790 to briefly summarize what we learned about combinatorics 4 00:00:08,790 --> 00:00:10,143 in this part of the course. 5 00:00:11,040 --> 00:00:14,280 Okay, we use permutations with variations 6 00:00:14,280 --> 00:00:16,920 when we must arrange a set of objects. 7 00:00:16,920 --> 00:00:21,000 In such cases, the order in which we pick them is crucial. 8 00:00:21,000 --> 00:00:22,860 The major difference between the two is 9 00:00:22,860 --> 00:00:24,240 that, in permutations, 10 00:00:24,240 --> 00:00:26,550 you always arrange the entire set of elements 11 00:00:26,550 --> 00:00:28,380 in the sample space. 12 00:00:28,380 --> 00:00:30,540 For instance, we would use permutations 13 00:00:30,540 --> 00:00:32,460 when we need to arrange the four runners 14 00:00:32,460 --> 00:00:35,130 we have already chosen for our relay. 15 00:00:35,130 --> 00:00:37,830 We have four runners and four positions. 16 00:00:37,830 --> 00:00:40,410 So we rely on permutations. 17 00:00:40,410 --> 00:00:43,230 If, however, we had to pick four out of six people 18 00:00:43,230 --> 00:00:44,063 on the team 19 00:00:44,063 --> 00:00:46,170 and then decide who runs which leg, 20 00:00:46,170 --> 00:00:48,960 we would require using variations. 21 00:00:48,960 --> 00:00:50,880 Alternatively, if we only care 22 00:00:50,880 --> 00:00:53,310 about which four out of the six runners made it 23 00:00:53,310 --> 00:00:54,360 into the team, 24 00:00:54,360 --> 00:00:56,790 we would be dealing with combinations. 25 00:00:56,790 --> 00:01:00,240 In this instance, we do not care who runs which leg. 26 00:01:00,240 --> 00:01:02,043 So order is irrelevant. 27 00:01:04,230 --> 00:01:05,519 Perfect. 28 00:01:05,519 --> 00:01:07,620 Another important topic we discussed is 29 00:01:07,620 --> 00:01:10,740 that there are two types of variations and combinations, 30 00:01:10,740 --> 00:01:13,290 with and without repetition. 31 00:01:13,290 --> 00:01:15,600 When we explore those without repetition, 32 00:01:15,600 --> 00:01:18,510 we see a clear relationship between permutations, 33 00:01:18,510 --> 00:01:20,883 variations, and combinations. 34 00:01:21,840 --> 00:01:25,500 The number of combinations equals the number of variations 35 00:01:25,500 --> 00:01:28,053 divided by the number of permutations. 36 00:01:29,040 --> 00:01:30,090 That is because we count 37 00:01:30,090 --> 00:01:32,580 all the permutations of a given set of numbers 38 00:01:32,580 --> 00:01:34,260 as a single combination, 39 00:01:34,260 --> 00:01:36,483 but as separate variations. 40 00:01:38,880 --> 00:01:42,960 We also define formulas we use to compute their values. 41 00:01:42,960 --> 00:01:44,133 P = n! 42 00:01:45,300 --> 00:01:49,768 V = n!/(n - p)! 43 00:01:49,768 --> 00:01:54,768 And C = n!/p! x (n - p)!. 44 00:01:56,880 --> 00:01:58,293 Recall that these formulas change 45 00:01:58,293 --> 00:02:00,960 when we include repeating values. 46 00:02:00,960 --> 00:02:04,290 V-bar = n to the power of p. 47 00:02:04,290 --> 00:02:07,223 And C-bar = (n + p - 1)!/p!(n-1)! 48 00:02:14,550 --> 00:02:15,810 Great. 49 00:02:15,810 --> 00:02:19,560 Furthermore, we talked about how combinations are symmetric. 50 00:02:19,560 --> 00:02:23,130 Choosing p-many elements out of a set of n-many elements 51 00:02:23,130 --> 00:02:25,140 can be done in the exact same way 52 00:02:25,140 --> 00:02:28,440 as choosing (n - p) many elements. 53 00:02:28,440 --> 00:02:30,750 The reason is we can reverse the problem 54 00:02:30,750 --> 00:02:33,333 and actually choose the elements to be omitted. 55 00:02:35,310 --> 00:02:36,270 So far, we've studied 56 00:02:36,270 --> 00:02:39,210 a significant amount of probability theory. 57 00:02:39,210 --> 00:02:40,230 In the next section, 58 00:02:40,230 --> 00:02:41,610 we will deepen our knowledge 59 00:02:41,610 --> 00:02:44,100 by learning how to deal with multiple events, 60 00:02:44,100 --> 00:02:46,320 introducing you to Bayesian Notation, 61 00:02:46,320 --> 00:02:48,270 independent and dependent events, 62 00:02:48,270 --> 00:02:51,810 and how to compute the probabilities for these events. 63 00:02:51,810 --> 00:02:52,920 See you there 64 00:02:52,920 --> 00:02:54,213 and thanks for watching. 4693