Combinations and Permutations Word Problems - Free Educational videos for Students in K-12

Combinations and Permutations Word Problems - By tecmath

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00:0-1	Good day . Welcome to take Math Channel . What
00:01	we're going to be having a look at in this
00:02	video is some combinations and permutations . Worded questions .
00:06	This is part of a series of videos . We've
00:08	been having a look at combinations and permutations . Uh
00:11	that is the number of different ways so that we
00:13	can select a number of items from a larger group
00:16	or even the number of different ways that we could
00:18	arrange things . So in this video I'm gonna chuck
00:22	six different combinations and permutations questions at you . I'm
00:25	not going to get right into uh necessarily going through
00:28	each rule and that sort of deal . I've done
00:30	that in previous videos , I'm just going to attack
00:32	each question . How could attack it and we'll go
00:35	with that . Okay , um anyway , six different
00:38	questions . See how you go . It's a question
00:41	one , a person has seven songs to choose from
00:44	and will perform three . How many different ways can
00:47	they do this ? So for any of these questions
00:50	, the very first thing you want to do is
00:52	really identified as a combinations question or a permutations question
00:56	once again with combinations order . Doesn't matter , say
01:01	like selecting three things from a bigger group , but
01:03	you don't really care what order they come out in
01:05	, but with permutations or it does matter . And
01:09	with this particular question it is a permutations question order
01:13	matters . There is a very definite first song ,
01:15	there is a very definite second song and there is
01:18	a very definite third song . So the word approached
01:21	this particular question is as follows , I would put
01:25	three different spaces for the three different songs are going
01:28	to be sung and then I go okay for the
01:31	first one , for the first particular song , how
01:33	many choices does this person have ? They have seven
01:36	songs they can choose from . And then for the
01:38	second song where they've already sung one of the songs
01:41	in this space here . So now they've only got
01:43	six spaces to choose from , six songs to choose
01:45	from . And for the third song , what ?
01:48	They've already sung two songs now from the seven ,
01:50	so now they only have five songs to choose from
01:53	. And what we then did all that we then
01:55	do is we multiply this through because it will work
01:58	out the number of different permutations . There are seven
02:01	times six times five is 210 different ways of selecting
02:10	these songs . Okay . You might have used the
02:13	rule for that and hopefully you would have got the
02:15	same answer for that . Okay , But that's the
02:17	way I attack these . We go to the second
02:19	question question two Horse race has 12 horses . How
02:24	many different ways can first ? 2nd , 3rd occur
02:28	. Okay , now , once again you're going to
02:30	look at this and say , well , is this
02:31	in this question ? Does order matter ? And I
02:35	would say it very definitely does . Because I've actually
02:37	quite specifically stated , we've actually we want a first
02:39	, a 2nd and 3rd . Okay , So how
02:42	would you do this ? You give it a fly
02:44	and you say you give it a go . Okay
02:46	. It's like you got the three different spaces ,
02:48	the first , the second and the third . And
02:50	for the first one , you have 12 possibilities ,
02:53	12 possibilities of which horse might come first . But
02:57	once that horse has come first , then you're left
02:59	with 11 different possibilities for second and then for third
03:03	, then you're only left with 10 different possibilities .
03:06	Okay , So what would you do then ? You
03:10	need ? Multiple this through 12 times 11 times 10
03:13	. So 12 , times 10 the answer is going
03:17	to be 1320 different ways of doing this , but
03:24	I'm just going to take this a little bit of
03:25	a further step . What if order ? Didn't matter
03:29	what if we were just saying , how many different
03:31	ways , you know ? Can we say the first
03:33	through what they throw us ? Three horses will be
03:35	without actually really caring whether their 1st , 2nd or
03:37	third . Well , we take this a little step
03:39	further and become accommodations question then and then what would
03:42	happen is we would say , well , okay ,
03:45	there's 123 different places here , and it means there
03:48	is three factorial ways of arranging these places . And
03:53	so we divide this out and that's how we do
03:54	combinations . Is we divide it by the number of
03:57	places here . Factorial list . Three factorial is three
04:03	times two times one , Which is equal to six
04:08	. So we're gonna end up with 13 , which
04:13	is going to be equal to 220 different ways and
04:17	that's if order , it doesn't matter . Okay .
04:21	And that's the extra step we take . We're doing
04:23	combinations questions . This is if you were just working
04:26	at the permutations and what it did matter , this
04:28	is if you're working in a number of different combinations
04:30	and order , didn't matter . So we go to
04:33	a different question Question three , How many different ways
04:36	can five cars be dealt from a deck of 52
04:39	cards ? Okay , now with this particular question is
04:43	all the matter . No , it doesn't because you
04:45	know the car that got dealt first could be dealt
04:47	fourth . So this is a combinations question . So
04:51	we're going to treat a lot . We did that
04:52	last part of that question . We did just before
04:55	, I'm going to get the fire spaces 123 For
05:00	five for the five cards . But then I'm going
05:03	to divide it By the number of different ways that
05:06	these five cards can be arranged . This is the
05:09	same as five factorial , five times four times three
05:14	times two times one . So for the very first
05:18	At the top of yeah , how many different ways
05:20	can 52 cards these ? We picked out ? Well
05:22	, the first one , we have 52 cards to
05:24	choose from . A second one , we have 51
05:26	cards to choose from the third card selection . Now
05:29	we have 50 cards to choose from the next card
05:32	. Now we have 49 cards to choose from .
05:35	And the next card . Now , Because we've already
05:37	chosen these ones , we have 48 cards and then
05:40	we'd multiply these through and you can imagine you're going
05:44	to get a pretty big number when you do this
05:46	. The number of different possibilities is 2590 . Okay
05:56	, Okay . The question for how many different ways
05:58	can the letters in the word Mississippi be arranged ?
06:02	Now this straightaway going to look at and say ,
06:05	is this a combinations or permutations question . And you
06:07	probably think what it does matter in this , what
06:09	it definitely matters . It's all about actually arranging things
06:12	in a particular order . So it's a permutations question
06:15	, but it's a bit of a special question because
06:18	It's not just a matter of arranging 11 different letters
06:21	, which there are in the word Mississippi because we
06:24	have these repeats . Uh if you were to change
06:26	the position of these peas , they're not going to
06:28	change the actual words . So we're going to take
06:30	into account these particular are repeats . So , the
06:34	way that we do this is as follows . First
06:38	off , the number of different ways that these letters
06:41	, Mississippi could be arranged is 11 factorial . Ok
06:45	, so 11 factorial Which means this 11 times 10
06:49	times nine times eight times seven times six times five
06:52	times four times three times two times one . But
06:56	to take into account that repeats , we just count
06:58	the number of seats for the different letters . So
07:00	for the letter He there is two of them for
07:05	the letter S there is four of these and for
07:08	the letter I there is four of these and we
07:11	divide these out as follows . So we're going to
07:13	divide by two factorial for factorial for factorial . So
07:19	we get something that looked like this . Now what
07:21	we can do is we could start counseling things out
07:24	this 4321 and this 4321 And you might say okay
07:28	this three and two here is equal to six and
07:32	we have a four and two which is equal to
07:34	eight . So I left with 11 times 10 times
07:38	nine times 7 times five . This is equal to
07:43	34,000 650 different ways . And that's the way you
07:49	do these types of questions . Okay , you just
07:51	divide by the repeats factorial list Question five . How
07:56	many ways can four fruits be selected from ? Six
07:59	for a salad ? Nice , easy question . Does
08:01	the order matter ? Hey , no , it doesn't
08:03	. This is a combinations question . This is a
08:05	really , really nice easy question . This one .
08:07	So I'm going to draw the four spaces for our
08:09	fruits 1234 And so for the first fruit we have
08:13	six to choose from . And now we've already chosen
08:15	that particular fruit . Now we have five . For
08:17	the second one we're going to select . And then
08:19	for the third fruit we're going to select . We've
08:21	already selected two out of the six . So we're
08:22	left with four . And for the next one we're
08:25	left with three . And we're going to multiply these
08:28	three . But with all combinations we have to divide
08:32	because we're gonna have to take into account the are
08:35	with it . It doesn't matter . We're going to
08:36	divide by the number of different ways . These four
08:39	objects can be arranged . Also to get rid of
08:42	those particular arrangements before 321 multiplied and that's what we're
08:49	left with so we can cancel out once again this
08:54	four is going to counsel this four out six is
08:56	going to be cancelled out by the three and the
08:58	23 times two is six and we're left with Five
09:02	times three which is equal to 15 different ways .
09:07	So for the 6th question , how many different ways
09:10	can six people sit around a campfire ? Does all
09:14	the matter in this question ? You might look and
09:16	say , hey , it definitely does . It definitely
09:17	does . So this is a permutations question . Okay
09:22	. But it's a special type of permutations question because
09:26	find themselves to draw this campfire and I'll draw the
09:29	six people sitting around it 123456 And you're going to
09:34	realize that these people are sitting in a circle .
09:36	And so for this particular question of these particular questions
09:40	, I'm going to consider that if everybody was to
09:44	get up and move counterclockwise this way and in particular
09:48	look at one person in particular , but you assume
09:50	that everybody's following his lead to this person will go
09:53	to where he is in , this person will go
09:55	to where he is and this person we go to
09:56	where he is . You're gonna notice that we're going
10:01	to assume that these are actually moving to a similar
10:04	sort of arrangement . Okay ? So if this person
10:08	was to move this way you'd have one similar arrangement
10:12	. Or if they were to move to this way
10:14	we could have another similar arrangement . Or if they
10:17	were all to get up and now move counterclockwise this
10:18	way that's still all be relative the same way .
10:21	Or they'll move this way we have a similar arrangement
10:24	again or they will to get up and move counterclockwise
10:26	. Once again they have a similar arrangement back to
10:29	the same position . We have 123456 . Similar arrangements
10:35	that we have to take into account . So we're
10:38	going to divide out by those . Okay ? So
10:40	we're gonna divide our answer by six . So six
10:44	different people . If they were in a line could
10:46	sit together six times five times four times three times
10:51	two times one Different ways . But these are going
10:55	to cancel each other out . So we're just left
10:57	with five times four times three times two times one
11:00	. Which is equal to 120 different ways . If
11:04	you were struggling with these , I recommend you go
11:06	have a look at a couple of combinations and permutations
11:08	videos . We were explained the rules in greater detail
11:11	. The methods were just use their in greater detail
11:14	. This was just a few practice questions . Just
11:16	mixing them up a little bit anyway . Um I
11:20	hope that video was of some help to you .
11:22	We'll see you next time . Bye .

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