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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