Combinations made easy - Free Educational videos for Students in K-12

Combinations made easy - By tecmath

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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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