Permutations and Combinations 1 (Counting principle) - Free Educational videos for Students in K-12

Permutations and Combinations 1 (Counting principle) - By tecmath

Transcript


00:0-1	Good day , Welcome to Tech Math channel . What
00:01	we're going to be having a look at in this
00:02	video is the counting principle and this is a way
00:06	of counting up various combinations really , really quickly .
00:09	So this is the start of a series of videos
00:11	. We're going to be looking at combinations and permutations
00:15	, which is pretty much a nice mathematical way of
00:17	saying , working on a number of different ways that
00:20	things can be arranged . For instance , how many
00:22	different ways , how many different combinations could you have
00:24	in a combination lock or how many different ways could
00:28	lotto numbers come out ? Or this one for instance
00:30	, say I had four books here , 1234 books
00:36	. And I was thinking to myself , how many
00:38	different ways can I arrange these on the bookshelves ?
00:40	So instant for instance , I could put them in
00:42	this order . The blue , black , red or
00:44	green or I could put them blue , black ,
00:47	green or red or I'm gonna take the red one
00:50	first and then the black one and then the green
00:53	one and then the blue one . And you might
00:56	then think to yourself , well how many different combinations
00:58	could you have ? And this is what this combinations
01:01	and permutations start to look at . And more specifically
01:03	today , this is what this counting principle will be
01:06	looking at . Okay , and we're going to be
01:08	using this in some later parts of what we're going
01:10	to be doing . So we're going to explain this
01:12	by going through an example and we just start by
01:14	imagining I'm choosing my outfit for the day . So
01:18	anyway I'll look in the cupboard and the first thing
01:20	I look down at is my shoes and I'm trying
01:23	to decide what shoes I have to wear . I
01:25	have black shoes and I have issues . Okay .
01:29	And then the next thing I'm gonna decide is what
01:31	color pants I'm gonna wear for that day . And
01:34	because I'm a funky sort of guy , I have
01:37	green pants and I also have orange pants . And
01:41	then the next thing and decide is what shirt I'm
01:45	going to wear . Okay , so I think to
01:47	myself , I have a red shirt , I have
01:51	a blue shirt and I also have a black shirt
01:55	. And so how many different combinations of outfits ?
01:58	How many different combinations of clothes can I wear out
02:01	of these shoes , pants and shirt combinations could I
02:04	make out of that ? So this is where we
02:07	use this counting principle to work out . But first
02:10	of what we're gonna do is we're gonna draw up
02:11	a tree diagram . Now a tree diagram pretty much
02:14	lists up all the different combinations that are possible .
02:17	So we look at all the different decisions that we
02:20	make . So the first decision we make is what
02:22	shoes I'm going to wear now , we could either
02:24	choose that . Are we wearing black or I'd be
02:27	wearing blue . So we have a different branch goes
02:30	off to each of those . The next thing we
02:32	decide is what color pants . So so for each
02:36	of these I might then decide I'm gonna wear either
02:40	green . So this one I could I could wear
02:42	black and then black shoes and then green pants or
02:45	blue shoes and then green pants . Or I could
02:48	use wear orange pants with either of these decisions .
02:54	The third thing I then do is look at shirts
02:56	, I could wear three different color shirts for all
02:59	of these . Yeah , so I could wear a
03:01	red shirt , Okay . Or I could wear you
03:04	guessed it blue or I could wear black . And
03:08	so what this tree diagram subject that lists off all
03:10	the combinations and what you'll see is if you count
03:12	the ends of the branches here , you see how
03:14	many different combinations we have . We have 123456789 10
03:20	11 , 12 combinations , 12 different combinations of outfits
03:25	I could wear . You're thinking to yourself , that's
03:28	that's pretty handy dandy . And you you look pretty
03:30	funky if you are very were various ones , but
03:34	you might then think is there a faster way doing
03:36	that without drawing out this tea tree diagram ? And
03:39	there is . And I'm gonna show you how to
03:41	do that because I have two pairs of shoes ,
03:43	two pairs of pants and three shirts . And a
03:47	very simple way of doing that is if I just
03:49	multiply these through two Times two is 4 times three
03:55	equals 12 . So say for instance , I are
03:58	very this up and all of a sudden now I
04:01	have six pairs of shoes , I have seven pairs
04:04	of pants and I have 12 pairs of shirts to
04:06	choose from . How would I go about rather than
04:08	drawing one of those tree diagrams which would take ages
04:10	and to be honest about it be rather squishy and
04:12	not very fun to draw . How would I go
04:14	about doing this ? Well , I just multiply this
04:17	through . So how many different outfits could I have
04:19	? six Times seven times 12 . Okay . So
04:24	that multiplied together . May I have 502 different combinations
04:31	of clothing I could wear . Okay , so that's
04:35	a really , really easy way of doing this .
04:37	Um And I could write out a rule for that
04:38	. But I think you really probably just get that
04:40	. You just get your number of different combinations and
04:42	then you multiply . So what about you try some
04:44	? So what about we have a menu ? Okay
04:46	you're going to restaurant . We have a menu .
04:48	And on this menu we have entrees and there are
04:52	four of these and then after the entrees you have
04:56	the Mains and on the main course there are 10
05:00	different main course dishes . And then on the desserts
05:05	we have three different types of desserts . And I
05:08	could ask you then how many different three course dinner
05:12	combinations could you make off this ? So I'll get
05:14	you to work it out . I reckon you need
05:16	probably I reckon you should have already worked it out
05:18	because I know you're all greater multiplying and I think
05:21	what you do is you go for Times 10 is
05:25	40 times three . Okay that's going to be 120
05:30	different combinations . Mhm . Okay . A bit of
05:34	a harder example now . So say we have a
05:36	lot of um where we have six numbers , so
05:39	six numbers and I'll draw them here . 123456 numbers
05:45	. And they get chosen from one balls numbered from
05:50	1 to 45 . Okay . And now I want
05:54	to know is how many different combinations do we have
05:57	here ? So this is a bit of a harder
05:59	one . And the way that we do this is
06:02	as follows . So I've got the slots here and
06:04	this is a really good thing to draw the start
06:05	. There's six different balls were choosing , I've drawn
06:08	six different slots , so there's gonna be 56 different
06:11	things were multiplying for the first slot here . The
06:13	first ball we choose out . Okay , we have
06:16	45 balls . 1 45 . You might think to
06:18	yourself well of the first ball we choose out .
06:20	We have 45 possibilities anywhere from 1 to 45 that
06:24	we choose for . The second ball . We choose
06:27	out . How many possibilities do we have ? Well
06:30	, one of these balls has been chosen out now
06:32	, so we only have 44 balls left . So
06:36	There'll be 44 different possibilities . Okay , What about
06:42	the next 1 ? Will have 43 possibilities . The
06:47	next one had 41 possibility because the ball is gone
06:50	. And then because that ball is going down the
06:51	next one , we have 40 possibilities . So how
06:53	many different combinations do we have ? And so you
06:58	might look at this thing . Okay , Okay .
07:01	How we work that out ? We're gonna multiply that's
07:02	a lot of numbers to multiply . We have the
07:04	number of 5,864 million 443 1002 100 combinations . So
07:18	one other one and this one , it's a bit
07:21	more of where I come from . I come from
07:22	victoria in Australia . And we have , well until
07:25	very recently we used to have license plates that were
07:27	like this , we would have three letters letter letter
07:33	letter and it was followed by three numbers . And
07:37	I've been in different places around the world and I've
07:40	seen that everyone has different types of our license plates
07:42	but this is the ones we have and you can
07:44	work this one out for your own license place where
07:46	you're from . So I want to know how many
07:49	different combinations we have that we could have here .
07:51	I mean obviously it's gonna be affected a bit .
07:53	Um they don't like to have rude words and that
07:55	sort of deal on license plates but we're not gonna
07:57	take that into consideration . You know we're not gonna
07:59	have P . 00 And poo or anything like that
08:02	. So how many different combinations license plates can we
08:05	have ? So you want things to yourself ? Okay
08:08	the first one , they could choose any letter .
08:10	There's 26 letters in the Alphabet . So there's 26
08:13	possibilities for that first letter . And then the next
08:16	one they could choose once again any letter because there's
08:19	still 26 letters . You could have a and then
08:21	you can have another a . So we're not worried
08:23	about repeats here . And then the next one ,
08:25	you can also choose 26 Letters and 26 letters .
08:29	Their numbers we have 0123456789 . That's 10 different possibilities
08:35	. 10 different and 10 different from the next ones
08:38	. So how do we work out ? How do
08:40	we count up all these combinations really quickly ? Well
08:43	we multiply them through . So 26 times 26 times
08:47	, 26 times 10 times 10 times 10 . And
08:50	the answer to this is there is 17,576 1000 combinations
08:59	. So you can work that out on a license
09:01	plate where you're from . Ok . And however they
09:03	work them out because they do vary . Uh huh
09:07	. But I'm just going to finish with one last
09:09	thing on this . Which is this idea to say
09:12	. What about it ? We're worried about a license
09:14	plate where we couldn't repeat any letters or numbers .
09:20	Okay . Every letter and number has to be unique
09:23	. How would you work that out differently ? It's
09:25	not a huge thing to do . Mhm Because of
09:28	the first number of letters that came out , you
09:30	would have 26 Now say it was any number one
09:34	of those letters that you took out , it would
09:36	only leave 25 letters For the next possibility and the
09:40	next one because now you've got rid of these two
09:42	would only lead 24 possibilities . Okay , so for
09:46	the numbers we have tend to start off with and
09:48	you guessed that the next one , we would only
09:50	have nine , the next one would only have a
09:53	and to work here . The number of combinations if
09:56	no repeats were available . So Would be to send
10:00	to multiply there and so you see that slight variance
10:02	there and that takes it down from 17 million to
10:06	11 million , 232,000 combinations . Anyway . Hopefully you
10:15	found this video informative . Um it's fairly intuitive ,
10:18	I find a lot of this stuff , but it
10:21	will get a little bit harder when we start to
10:23	get into the harder combinations and permutations . And I'm
10:26	sure some people are going to come out with some
10:27	really gnarly sort of uh comments and questions . So
10:32	anyway , hope you found that good . See you
10:34	next time . Bye .

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Permutations and Combinations 1 (Counting principle) is a free educational video by tecmath.

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