Permutations and Combinations - Permutions. - Free Educational videos for Students in K-12

Permutations and Combinations - Permutions. - By tecmath

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00:0-1	Good day . Welcome to the tech mouth channel .
00:01	What we're going to be having a look at this
00:03	video is permutations . Permutations . The number of different
00:07	ways that things can be arranged . This is not
00:09	to be confused with combinations . Are we trying to
00:11	actually address right now ? First off , the difference
00:14	between combinations and permutations And I'll explain explain this with
00:18	an example . Imagine we had 20 people . Okay
00:23	20 people and I have three seats . I'm going
00:26	to place them in Now . I select out of
00:29	20 people . I select three people and one is
00:32	Raj . Another one is die and another one is
00:38	ty So you're always talking about permutations where I placed
00:44	raj die and tie would actually matter because it would
00:48	be considered different . That raj would sit here ,
00:50	die would sit here . Ty would sit here as
00:52	opposed to die . Sitting here raj sitting here ,
00:55	tie sitting here or tie . Sitting here , Die
00:59	sitting there and ranching in there . That would be
01:00	considered different arrangements . Whereas we're talking combinations arrangement doesn't
01:05	matter . And about 20 people . I've just selected
01:07	three people and it happens to be these three .
01:09	And so combinations wouldn't really matter where these three guys
01:13	sat as long as they were sitting somewhere . So
01:15	I hope you get that idea . But in this
01:17	particular video where it is going to be considering permutations
01:21	so order does matter . So there's four major different
01:25	types of permutations that we're going to be having a
01:28	look at in this video . We're going to start
01:30	simply and we're going to work our way up as
01:32	we go . All right . So I'll recommend if
01:34	you're doing permutations as a topic , you probably will
01:37	want to have a look at this entire video .
01:39	So first off I'm going to start with just examples
01:42	with these . So how many different ways this ,
01:45	This is the most simple type of permutation . How
01:47	many different ways can we arrange for people in a
01:51	row ? So to do this , you might want
01:54	to consider . We have four different positions . Okay
01:58	, so four people , four different positions . And
02:01	how many different ways could we arrange them ? Well
02:04	, if we consider this first position , we have
02:07	four different people , we could choose to be in
02:08	here . But once we've placed person in here ,
02:11	we left . Now with only three people for the
02:13	second position . Okay , so now we've placed these
02:16	two people in were narrowly left with two people and
02:19	then we're only left with one person . So when
02:21	we're talking about how many permutations we have , what
02:25	we do is we then multiply these through . So
02:27	four times three times two times one . And if
02:30	you've had a look at my earlier video , the
02:32	video I made just before this on factorial , you
02:34	realize this is four factorial four times three times two
02:37	times one is equal to 24 different arrangements that we
02:42	could have . So that's the first top of permutation
02:45	we can have . I'll tell you what , I'll
02:47	give you an example that you can do . So
02:49	what about imagine if I was talking now about what
02:53	about six people in a row ? How many different
02:55	ways can we arrange six different people in a row
02:58	? So this is equal to six factorial , which
03:02	is an extra two positions . So this is six
03:06	people 54321 And we multiply these . And that's the
03:13	simplest way of doing that . If you were to
03:15	multiply this out , you're going to get 720 different
03:19	ways . Okay , so I'm going to go through
03:21	the second type of permutation now that we get the
03:24	second most common type where what we have to say
03:27	something like so imagine we had seven books but we
03:30	only had three spaces to put them in . Okay
03:34	. On the bookshelf it's only space for three books
03:36	. We have seven books to choose from how many
03:38	different arrangements would be possible . So I'm going to
03:41	draw those three spaces 12 three . Now , if
03:46	you imagine this from these seven books which you have
03:49	to choose a book but there's seven possibilities we could
03:52	choose from . So after we choose that first book
03:55	we now have one less and we've only got six
03:58	possibilities to choose from . Then we only have five
04:00	possibilities to choose from . After we've already filled these
04:02	two positions . And this is the number of arrangements
04:06	we could have . We would then get these and
04:07	multiply them . We're not going to multiply the entire
04:10	factorial because we only have three spaces , so seven
04:14	times six times five Is 210 different arrangements . Okay
04:22	. Yeah , I'm just gonna slow down on this
04:24	one and show you something with this . So just
04:26	to give you a bit of a hand in case
04:27	you get these types of questions . So what you
04:30	might notice with this is that this seven times six
04:34	times five is kind of like seven factorial but we're
04:37	missing the last little part . We're missing this four
04:40	times three times two times one part . Okay .
04:44	This part . Mhm . We're missing that . And
04:48	what your might also realizes with this , where are
04:52	they taking the first three spaces ? Because we only
04:55	have three spaces . So this has led to a
04:57	rule that if we have n number of items and
05:00	we have this many spaces , the number of different
05:03	ways they can be arranged is where we have N
05:06	factorial over n minus R . Factorial . Okay .
05:11	I'm pretty much what we're doing is we're saying where
05:13	are we going to take the first this many spices
05:18	? Okay . So what about I'll give you an
05:19	example here . What about we have eight cars and
05:23	we have three spaces to park these cars . All
05:28	right . How many different ways could this be achieved
05:32	or arranged ? So first off , what you're going
05:35	to do is you're gonna probably think , okay ,
05:36	we're gonna get eight factorial different ways but we're only
05:40	going to take the first three . Okay . Because
05:48	we only got three spaces . So this is this
05:51	is the way I think about it . It's just
05:53	going to be eight times seven times six . Okay
05:57	. Which is 336 different arrangements . So I actually
06:04	prefer to think about that way rather than actually thinking
06:08	about some rule , The next type of permutations we're
06:11	going to consider are ones that involve repeats . And
06:14	I'll give you an example of this . They asked
06:17	to write the word puppy down and then I want
06:19	to ask you how many different ways can we arrange
06:23	these letters ? Now ? This is not just a
06:25	simple matter of just arranging five different letters because you're
06:28	going to notice that a couple of letters are actually
06:31	the same P and P and P . We have
06:34	three different piece . So we could swap these two
06:36	piece here and we would end up with the same
06:38	combination . We can also swap these piece here and
06:41	end up the same combination . Would still say puppy
06:43	or these two would still say our puppy . So
06:46	how many different ways can we arrange these letters taking
06:50	into account the repeats . So it's a fairly simple
06:54	way of doing this . But we start out with
06:55	this just having a look at how many letters in
06:58	total there are . So there's five letters in total
07:00	. So if we were just to consider this where
07:02	they were different , all the letters were different .
07:04	We would have five factorial different ways of arranging these
07:08	five times four times three times two times one ,
07:12	Which is 120 different ways of arranging these . But
07:17	considering this letter P here , we have This letter
07:21	P Occurring three times . Okay . And this letter
07:25	P could be arranged how many different times where you
07:27	might then even trying to arrange these , but you
07:29	might even think they can be arranged three factorial ways
07:33	. So , when we're trying to figure out how
07:34	many different ways the letters puppy can be arranged .
07:36	We start with five factorial and we divided by three
07:40	factorial . Ok , So we'll start with a number
07:43	of different letters altogether , in the number of repeats
07:45	. And how many ways they could be arranged ?
07:47	And we divide it out . So three factorial is
07:51	equal to 63 times two times one . What ?
07:54	We end up with 120 divided by six . We
07:58	have 20 different arrangement . So I hope you get
08:03	that idea . Okay , all you need to do
08:04	with that is you divide out the repeats factorial list
08:08	. So say for instance , I now consider a
08:10	different word . I'm going to consider the word mammal
08:14	and I'm gonna get you possibly if you want to
08:16	try and actually work and how many different ways we
08:18	can arrange these letters . Uh Now first off ,
08:22	I'm going to start answering this right now . You
08:23	can pause it and give it a go . But
08:25	we have six factorial different ways that we could arrange
08:29	these letters that we don't consider repeats . Six factorial
08:32	is six times five times four times three times two
08:36	times one , which is 720 . We have two
08:40	different types of repeats . Here we have the ems
08:43	the ems , we have three of these and we're
08:46	eyes , we have two of these . So we're
08:48	gonna divide by three factorial . Two factorial . Ok
08:55	, so three factorial three times two times one .
08:59	I'm not gonna bother right the times one because we
09:01	all know that's not going to change the result .
09:03	And this one here is two times one . So
09:05	we're going to times up to as well . So
09:07	three times two is six times two is 12 .
09:12	So how many different ways can we arrange these ?
09:14	720 divided by 12 . We have 60 different ways
09:20	. So hopefully you're getting that idea . Okay ,
09:23	all you need to do is you need to work
09:24	out the repeats , the new factorial is them and
09:26	divide them through . It's a nice way of thinking
09:29	about it . I kind of make sense when you
09:30	think about it as well . So the last type
09:33	of question that you're going to get with these is
09:36	where we are going to consider permutations that you have
09:40	when you have a circle . Okay , so how
09:42	many different ways ? Imagine this sort of question .
09:45	We have four different people , but we're trying to
09:47	arrange them in a circle . How many different ways
09:50	can we do this ? I'll tell you what ,
09:51	I'll draw this circle down here . And I'm going
09:52	to explain the problem that we get with this and
09:56	then it will lead to our solution . So imagine
09:58	how four different people are going to call them ,
10:00	I B c D and we arrange them . But
10:06	then you're going to notice that we can also arrange
10:08	these people . Well , we could put a here
10:13	be here see here D . Here or a here
10:17	be here see here D . Here or a here
10:21	be here . See here D here you notice that
10:25	all we're really doing is we're just rotating . It's
10:27	exactly the same . The arrangements the same . Now
10:31	look if you were actually saying , hey , no
10:32	way . So it's not , what you can do
10:34	is you can just treat this as four factor of
10:36	the exact position . These guys said matters treated as
10:40	four factorial . It's like arranging them in a line
10:43	. But if all of a sudden you say ,
10:44	hey , I actually don't mind because these people counterclockwise
10:48	or clockwise are sitting relative to each other in the
10:50	same position . You know , the rotation can be
10:55	taken into consideration here . What we need to do
10:57	is we just have a look together 1234 rotations .
11:02	So what we actually do is we defined by these
11:05	number of rotations . Okay , so four factorial divided
11:09	by 4 to 4 factorial is four times three times
11:13	two times one , divided by four . Well we're
11:17	just going to counsel these out and what we're actually
11:19	left with is four take away one factorial . In
11:23	fact , the way that we work this out ,
11:24	the formula in a circle is n take away one
11:28	factorial . That's the way you work out the number
11:31	of permutations in a circle , which is going to
11:33	be three times two times one , which is six
11:37	. So what about you if I was to ask
11:38	you , how many different ways could we arrange ?
11:43	What about six people in a circle ? Okay ,
11:47	so six people in a circle ? Six people in
11:50	a circle and you're just gonna go , okay ,
11:52	six take away one is five . Factorial is going
11:55	to be five times four times three times two times
11:59	one , which is going to be 120 . So
12:04	there are different types of questions you get with these
12:06	. They're fairly simple and they make really good sense
12:09	when you think about them . Anyway . Thank you
12:11	for watching . Hope you found this video informative .
12:14	Uh , see you next time . Bye .

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