Combinations made easy - By tecmath
Transcript
00:0-1 | Good day , Welcome to Tech mount Channel . What | |
00:01 | we're going to be having to get in this video | |
00:03 | is how to work at the amount of combinations possible | |
00:06 | when we select a number of objects from a larger | |
00:08 | group . This is part of a series of videos | |
00:10 | . We've been looking at combinations and permutations . So | |
00:14 | the first thing I to address is how combinations different | |
00:16 | from permutations , which is this with combinations or it | |
00:20 | doesn't matter . Let's give an example of this . | |
00:23 | So sales had five books , five different books and | |
00:26 | I'll write that down and I was going to select | |
00:30 | three of them . Yeah , I was going to | |
00:32 | select these three books and I'm going to take them | |
00:34 | on holiday show you the books here . I've got | |
00:36 | a greenie sort of color book . I got a | |
00:38 | dark green sort of book . I've got a black | |
00:40 | book . I've got a blue book and I have | |
00:43 | a red book . And from these five books , | |
00:45 | I'm going to select three of them to take away | |
00:47 | . But it doesn't matter what order my grandma was | |
00:49 | going to chuck them in the suitcase with combinations . | |
00:52 | Or it doesn't matter how do I go about working | |
00:54 | ? How many different ways I could do this ? | |
00:57 | Well the first thing I do is I'll show you | |
01:00 | the spaces for three books . 123 And what you | |
01:03 | might realize is when I select my first book , | |
01:05 | I've got five different . This is the first one | |
01:07 | to go here . I've got five books to choose | |
01:09 | from 12345 So I want to put that number in | |
01:11 | there . So I select the first book . So | |
01:14 | it's a red one there . And it leaves me | |
01:16 | with now four books to choose from for this second | |
01:19 | space . Maybe this book . And then it leaves | |
01:22 | me with three different books for the third space . | |
01:25 | So all we need to do now to work out | |
01:27 | the number of different ways that I can get these | |
01:30 | books are different . This is going to be for | |
01:32 | permutations and I'll show you how to extend this for | |
01:34 | combinations is we just multiply these through five times four | |
01:38 | times three , which is equal to 60 . There's | |
01:41 | 60 different ways . This can be done what this | |
01:43 | is showing us at the moment , it's saying that | |
01:46 | pretty much that read and if I had chosen a | |
01:49 | blue and this green here that it would be considered | |
01:52 | different to green , blue and red . But you're | |
01:56 | going to see this is just the same books would | |
01:58 | swapped around in order . So permutations would include this | |
02:02 | sort of thing and combinations would say , well hang | |
02:04 | on now , these are the same , they're just | |
02:06 | in a different order . So this is permutations are | |
02:09 | answer here , we've got 60 . So we just | |
02:10 | need to do one extra step in order to work | |
02:13 | out how many combinations of things we can choose . | |
02:16 | And that's fairly logical when you think about it . | |
02:19 | Because if we look at our three spaces here , | |
02:21 | you might say well how many different ways can these | |
02:24 | be ordered ? How many different ways can three spaces | |
02:26 | be ordered ? And you might remember from a few | |
02:30 | other videos , we have three spaces here to choose | |
02:32 | from to here and one . They had this many | |
02:36 | different ways that three different objects can be ordered . | |
02:39 | Okay , the three different spaces . So this is | |
02:41 | written in quite often as three factorial three times two | |
02:45 | times one , which is six different ways . And | |
02:47 | we divide this through and this will tell us the | |
02:50 | number of combinations as opposed to the number of permutations | |
02:54 | , 60 divided by six equals 10 . I'm going | |
02:58 | to show you this now using the rule that they | |
03:01 | actually write down . But I'll tell you the truth | |
03:02 | , I actually don't use the rule very much . | |
03:04 | They are combinations rule but it's a handy one to | |
03:07 | wear to work through just in order to get a | |
03:09 | bit of understanding , but that's the way we work | |
03:11 | out combinations . So first off , this is the | |
03:14 | rule and I'm going to show it as we go | |
03:16 | with our example . So say we were looking at | |
03:19 | combinations and from five things , we were going to | |
03:23 | slip through them the way they right . This is | |
03:25 | as follows , they're right combinations and from n objects | |
03:31 | with selecting are different things . Okay . And this | |
03:35 | equals this one equals Well , what we did is | |
03:39 | we selected three things . Okay , But I'm going | |
03:42 | to write this down a little bit weirdly , this | |
03:44 | is five times four times three times two times one | |
03:50 | because I'm dealing with the numbers we have here to | |
03:52 | help us work out our rule . But what we | |
03:54 | actually we're only left with was the first three here | |
03:57 | . Okay . I'm going to notice we didn't take | |
03:59 | the end to in fact what we didn't take was | |
04:01 | this two times one part ? This part . Okay | |
04:04 | . I just got a little line from just to | |
04:05 | show that we didn't take them . And on a | |
04:09 | rule this could be written as N factorial . A | |
04:12 | factorial . The number of times the most live factorial | |
04:15 | is five times four times three times two times 13 | |
04:18 | factorial with three times two times one . This is | |
04:21 | N factorial over And this is N . Take away | |
04:26 | our factorial it and take away our It's two factorial | |
04:31 | . Ok . But then we just have one little | |
04:34 | extra thing we've been divided by . And you remember | |
04:36 | that we had the number of spaces . This number | |
04:38 | here they are . Okay , this was three times | |
04:42 | two times one . Which is r factorial . Okay | |
04:47 | , that was what we divided by . So you | |
04:49 | might look at this and say OK , this was | |
04:51 | five factorial . Divided by two factorial and take away | |
04:57 | our factorial divided by three factorial . Okay . Also | |
05:02 | divided by three factors . And that's the rule we | |
05:04 | used but I don't particularly years that I lived like | |
05:06 | that . I tend to draw it out like I | |
05:07 | did when I R . Was working it out before | |
05:10 | . So we'll go through a couple of other examples | |
05:12 | . For instance , we had a committee of four | |
05:15 | people . Okay , We had four people that we | |
05:17 | were going to select from a bigger committee offense . | |
05:22 | So four people selected from 10 . And how many | |
05:28 | different ways could we do this ? And this is | |
05:30 | where we don't particularly care about order . So this | |
05:33 | is going to be how many different combinations are possible | |
05:36 | . So you might just get used to this , | |
05:39 | you might write their combinations and you might say 10 | |
05:43 | and four from 10 people . They're going to select | |
05:45 | four positions in this equals . And so we're going | |
05:49 | to have those four positions . Those 1234 positions . | |
05:54 | Okay , So the first one is tend to choose | |
05:56 | from 10 people and then that positions taken . So | |
05:59 | we only have nine people to choose from there . | |
06:01 | Now we got two positions taken . So only leaves | |
06:04 | us with eight people to put in that position and | |
06:06 | then there's three people here . So this only leaves | |
06:08 | seven people and then we're going to end up multiplying | |
06:12 | these . Okay , this is n . Factorial over | |
06:16 | and take away our factorial part of it . But | |
06:19 | then we're going to divide this by you remember the | |
06:22 | number of different spaces here . Factorial is so over | |
06:25 | four times three times two times one . And I'm | |
06:30 | just going to fill that in and we're going to | |
06:33 | get what our answer is . You can do this | |
06:35 | using a rule if you want , but I just | |
06:36 | do it this way . So I can cancel out | |
06:39 | because eight is the same as four times two . | |
06:42 | And then I can also say , okay , well | |
06:45 | this is three and I'm gonna divide this by three | |
06:47 | and get three . So 10 times three is 30 | |
06:50 | times seven is 210 over one to answer . There's | |
06:55 | 210 different ways of doing this . What about one | |
06:59 | last one of these ? Now , the question I | |
07:01 | have for this one is , how many different ways | |
07:04 | can we select from eight people ? So from eight | |
07:06 | people , we're going to select five kids to play | |
07:13 | basketball . Okay , So how many different ways can | |
07:15 | we do this ? So you might give this a | |
07:17 | go . All right now , how would I do | |
07:20 | this ? First off I'm going to have five different | |
07:23 | spaces . 12345 And this is going to be filled | |
07:27 | from these eight people . This eight by seven by | |
07:29 | six by five by four . Okay . And it's | |
07:33 | the same sort of explanation we're using Now . We | |
07:36 | had five different spots . So this is gonna be | |
07:38 | over five times four times three times two times one | |
07:43 | . And we can now start canceling out . We | |
07:46 | have a five here and a five here . I | |
07:48 | can put little multiplication is here actually . All right | |
07:52 | . We have a four here and of four here | |
07:57 | . Two times three is the same as six . | |
08:00 | This leaves us with a one down the bottom , | |
08:02 | which is not going to really mean that much . | |
08:04 | And so seven times eight equals 56 . And that's | |
08:09 | the way you work out combinations . It's pretty crazy | |
08:12 | right ? It's pretty easy . Um Anyway hopefully you | |
08:16 | get this idea . In fact if I was going | |
08:18 | to write it's just before I go you're going to | |
08:19 | write this factorial and using the rule I'll just quickly | |
08:21 | do that for you . Um What about I jot | |
08:24 | that down , what would you call that do you | |
08:26 | think we'd call this eight factorial over N . Minus | |
08:31 | R . Which is um you know that eight take | |
08:34 | away five which is three factorial over five factorial . | |
08:39 | All right it's pretty simple . Right ? Anyway hopefully | |
08:42 | that video is of great help to you and oh | |
08:46 | so you next time . Bye . |
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