00:00Hi guys, okay, so this
00:02is the last lesson on
00:03counting principles and These are
00:07the hardest types of questions.
00:08This is keeping objects together
00:10are separated. So I think
00:11I said in the Opening
00:14lesson I said if you
00:15can find if if you
00:16wanted to find a really
00:17challenging question that I couldn't
00:19do these are the types
00:20To look focused some of
00:22them are just so confusing.
00:23Anyway, let's let's go through
00:25them what we're looking at
00:26is so we're we've to
00:27keep objects together
00:28together are separated. Here, for
00:29example, look at this example,
00:31two eight boys and two
00:32girls on a bench. How
00:33many arrangements are there given
00:35that the girls do not
00:36sit together or the girls
00:37have to sit together or
00:38something like that? I'm going
00:39to start though with this
00:45question. How many permutations, so
00:47let's arrangements, of the word
00:49triangle have the letters n
00:51and g next to each
00:52other. Okay. This is how
00:55you do it.
00:56You treat N and G
00:59as one thing. So I
01:00look for many of these
01:02types of questions. I like
01:03to just, I like to,
01:04I like boxes. I like
01:05to write out boxes like
01:07this. So one, two, three,
01:08four, five, six, seven, eight.
01:10There's eight letters. So I'm
01:11going to do one, two,
01:13three, four, five, six. And
01:17then instead of seven and
01:19eight, I'm going to do
01:20something like this. So these
01:21two boxes that are N
01:22and G are kind of
01:24going together. And we're going
01:25to treat them, you're going
01:27to treat them as one
01:28object. But because there's two
01:30letters, we'll just, whatever answer
01:32we get at the end,
01:32we'll just multiply by two.
01:35So the number of permutations
01:37is just so how many
01:38objects do we have? We've
01:39one, two, three, four, five,
01:40six, seven objects. So that's
01:42just seven factorial. If I
01:44want to find out how
01:45many ways can I arrange
01:46these letters, seven factorial? By
01:50the way, all the letters
01:50are different here.
01:52Well, maybe I will show
01:54you how to do on
01:55where the letters are the
01:56same, but it specifically says
01:58in the guide that you
01:59don't need to know how
02:01to do that. So I'll
02:02just mention it after this
02:03example, but note you don't
02:06need to do arrangements where
02:11two objects are identical. That
02:14specifically says that in the
02:15good. Okay, so this is
02:17some factorial and then I
02:19just multiply it by two
02:20because think about it, it's
02:23all the permutations where I've
02:25got n and g, but
02:28I've those exact same permutations
02:30where I have g and
02:31n, which are obviously different.
02:32So I just multiply it
02:33by two. If I had
02:35to keep three letters together,
02:36I'd multiply it by three
02:37factorial because you'd kind of
02:39permute the letters that are
02:43together. So zen factorial times
02:46two, obviously, I don't know
02:48how to do that with
02:48my
02:48A calculator, seven factorial times
02:52two is 10 ,080, 10
03:00,080. That's how many different
03:04arrangements of the word triangle.
03:08Now, just to show you
03:13how to do this, imagine
03:14I had,
03:16Let's say imagine I had
03:18the word, let's think of
03:21a word that does have
03:23two letters like for example,
03:26spoon. Don't ask me why
03:28I thought of that word,
03:29but spoon. Imagine I have
03:30the word spoon. How many
03:32ways can I arrange these
03:35letters? Now note the two
03:37o's or how many words,
03:40well not words, but how
03:41many ways can I arrange
03:42the letters?
03:44Because these are identical, the
03:50arrangement spoon is exactly the
03:55same as if I range
03:56it like this SPOO and
03:58but I swap these two
03:59O's. So it's the same
04:00thing. So what you actually
04:02do is you say it's
04:06five factorial, five factorial, and
04:09then you divide by two
04:11factorial.
04:12However, many of these arrangements
04:14there are, I need to
04:16get rid of them. So
04:16I divide by 2 factorial.
04:19That will give me, well,
04:21whatever that gives me, that's
04:22120 divided by 2 is
04:2460. So there's 60 ways
04:25you can arrange that. So
04:27if you have two identical
04:29objects, you divide by 2
04:30factorial. But as I said,
04:31you don't need to know
04:32how to do that, specifically
04:34says that. So even enough
04:38to know without having to
04:40worry about that.
04:40Okay, part B. How many
04:45permutations of the word triangle
04:46have none of the vowels
04:47together, right? So I find
04:50this the more difficult one.
04:51When you have to keep
04:52things separate. So I've titled
04:56this, keeping objects together or
04:58separated. So here we have
05:00to keep the vowels. Sorry,
05:03none of the vowels can
05:04be together, so we have
05:05to keep them separated. Now,
05:07the way we do it
05:08is this.
05:08Let's look at the non
05:11-voles, the consonants. So I
05:13have 1, 2, 3, 4,
05:165, 5 consonants. So I'm
05:18going to write them down
05:19with the boxes again. 1,
05:212, 3, 4, 5. Now,
05:28those, I can permute them
05:30if you like, so that's
05:325 factorial, because I can
05:33write them in 5 factorial
05:36different arrangements.
05:36Now the three vowels that
05:41I, A and E, where
05:44can they go? Well, if
05:46they have to be separated,
05:47they have to go into
05:48the gaps here. So they
05:49have to go in here
05:50or here, or here, or
05:53here, or here. So they
05:55have to go in the
05:56gaps, I'll call those gaps.
05:58And where can I put
06:00them? Well, I could put
06:00I there, A there and
06:02E there, or I there
06:03and A there and E
06:04there.
06:04are either and either and
06:06either or e a I
06:08mean there's there's loads of
06:10different combinations. So how do
06:11I figure out how many
06:12combinations there are? Well there's
06:14one two three four five
06:15six gaps and three letters.
06:19So that's just six choose
06:21three six choose three and
06:25because the three can be
06:30in
06:33order does matter here. So
06:34i a e is not
06:36the same as a e
06:38i. I have to multiply
06:39by three -factor. Now I
06:41could have just said six
06:42p three. But I think
06:45this is better for understanding.
06:48So that leaves me with
06:52five -factororial, five -factororial multiplied
06:57by probability
07:01probability combinations 6, 3 and
07:08then multiply by 3, factorial
07:12gives me 144, 0, 0,
07:161, 4, 4, 0, 0,
07:20different permutations of the word
07:21triangle where these the vowels
07:24have to be kept separate.
07:28Okay.
07:29Example two. This is straight
07:30from a past paper question.
07:33Eight boys, two girls sit
07:35on a bench to determine
07:36the number of possible arrangements,
07:39possible arrangements, given that the
07:44girls do not sit together.
07:46Okay, so let's do that
07:49the same way. So eight
07:49boys, two girls. So same
07:51thing. I have one, two,
07:53three, four, five, six, six,
07:57seven eight boys to where
07:59can the girls go? Well,
08:00they can either go here
08:02here here here here here.
08:03How many gaps are there?
08:05Well, there's nine one two
08:06three four five six seven
08:08eight nine. There's nine gaps.
08:09So I'm going to do
08:10my eight factorial. That's that's
08:13the just the boys. The
08:15boys can be ordered in
08:16eight factorial different ways. Then
08:18I'm going to do nine
08:20choose two because there's nine
08:23gaps. I need to put
08:24two girls
08:25into the gaps and then
08:28I need to do two
08:29factorial, same reason as above,
08:32because the girls can go
08:34here and here is different
08:36to here and here if
08:37you swap the girls around,
08:39so that's why you don't
08:39want to work with two
08:40factorial and this gives me
08:43eight factorial times
08:53nine comma two times again
09:00yes you could have just
09:01done np2 here and pr
09:06nine comma two would would
09:07work fine but I actually
09:10think as I said helps
09:11with understanding if you did
09:12this way okay so look
09:14a lot of combinations this
09:16is two million nine hundred
09:17three thousand and forty two
09:18million nine hundred and three
09:20thousand and
09:2140, part B. Now the
09:25girls do not sit at
09:27either end. Okay, eight boys,
09:31two girls. So there's 10,
09:3210 people, one, two, three,
09:35four, five, six, seven, eight,
09:40nine, 10. So the girls
09:43are not allowed to sit
09:44on either end. So this
09:44has to be a boy,
09:46and this has to be
09:47a boy. So see what
09:48I mean by right?
09:49find just writing out these
09:50kind of block boxes, you
09:52can at least, you can
09:54get some kind of visual
09:55to what's going on. So
09:57I have to have a
09:58boy at either and I
09:59have a boy at either
10:00end. How many boys can
10:01go here? Well, there's only
10:02eight boys. So eight. And
10:04then if I put one
10:05there, how many boys are
10:06left here? Seven. So this
10:08is eight times seven. Now
10:13eight times seven, and I
10:16have one, two, three, four,
10:17or 5, 6, 7, 8
10:19people left and they can
10:20go in any order. So
10:22it's just eight factorial. There's
10:24no restraints there. You can
10:27have a girl, you can
10:27have a girl, a girl,
10:28they can sit together, you
10:29can have boy, boy, girl,
10:31boy, girl, whatever. They can
10:32go in any different order.
10:33So it's just eight factorial.
10:34So eight times seven times
10:36eight factorial is eight times
10:40seven times eight factorial.
10:452, 2, 5, 7, 9,
10:482, 0, 2, 2, 5,
10:507, 9, 2, 0, 2,
10:535, 7, 9, 2, 2,
10:545, 7, 9, 2, 0,
10:57correct, right, last one. Last
11:01one. The girls do not
11:03sit on either end and
11:05do not sit together right.
11:06So look, this is, this
11:09is bad as it gets,
11:09I think in my opinion,
11:10this is, it's kind of
11:11a mix between the two.
11:12And I think if you,
11:13if you understand
11:13done these two, you should
11:15be okay with this. So
11:16there's, right, again, we're gonna
11:20have a boy and a
11:23boy at either end here,
11:27boy and a boy. But
11:28now the girls do not,
11:29they do not sit on
11:30either end fine and they
11:31do not sit together. So
11:33firstly, this is gonna be,
11:34this is gonna be eight
11:35times seven, because I have
11:37eight boys here and seven
11:39boys here. So eight times
11:40seven fine.
11:41Then it says, the girls
11:44do not sit together. So
11:45how many boys do I
11:46have left? Well, I have
11:47six boys left. One, two,
11:50three, four, five, six, six
11:55boys left. And then the
11:56two girls can again go
11:58into the gaps. Now how
11:59many gaps do I have?
12:00One, two, three, four, five,
12:03six, seven. So I need
12:05to put the two girls
12:08into those gaps.
12:09So I have to choose
12:11two gaps out of seven.
12:13So seven choose two. Again,
12:17I can, the girls can
12:18go, you can swap them.
12:23So order matters. So I'm
12:24going to multiply by two
12:25factorial again, because if I,
12:28if I've, if I've girl
12:29a here and girl b
12:30here, well, it's a different
12:32order. If I put girl
12:33b here and girl a
12:34there. And then finally,
12:37Finally, don't forget that these
12:406, 1, 2, 3, 4,
12:415, 6, these 6 boys,
12:43they need to all be
12:45permuted, except that word, which
12:49is 6 factorial, because these
12:51are going 6 factorial different
12:52ways. So yeah, I know
12:55that is complicated. And that
12:56equals, let's do this. So
13:00I have 8, 8 times
13:037,
13:06times, I'm going to do
13:08my factorials first. Now let's
13:10do it how I see
13:11it. 8 times 7 times
13:13probability combinations 7, 2, 7,
13:212 times 2 factorial, which
13:27is just 2 but whatever,
13:29and then times 6 factorial,
13:32and that gives me 1
13:34six nine three four four
13:35oh one six nine three
13:39four four oh okay um
13:43yeah look that's gonna take
13:46you a while to get
13:46your head around because honestly
13:48when I first saw that
13:49question it took me a
13:51long time to get my
13:52head around it especially as
13:55I said I find the
13:57ones where where you've to
13:59keep them separated the most
14:00tricky but if you write
14:02If you actually write out
14:03like this and look at
14:04the look for the gaps,
14:06where can they go? I
14:08think that really helps figuring
14:10it out. This example here,
14:14look, it's certainly worth, it
14:16was definitely worth me mentioning
14:18it. It's worth me mentioning
14:20and it's something you can
14:22know, but you don't have
14:23to know it. You won't
14:24be asked, you won't be
14:25asked this in an exam.
14:27Okay. I will see you.
14:30in the next lesson, we're
14:31finished counting principles. So obviously
14:35go away and practice as
14:36many of these types of
14:37questions as you can. And
14:38I personally am moving on
14:40to complex numbers. You may
14:42be wherever you are. And
14:45I'll see you in that
14:47video. Good luck.