• Congratulations to the Class of 2024 on your results!
    Let us know how you went here
    Got a question about your uni preferences? Ask us here

Perfecting Permutations and Combinations (1 Viewer)

Sy123

This too shall pass
Joined
Nov 6, 2011
Messages
3,730
Gender
Male
HSC
2013
So I have come to the realisation that the only thing weighing me down at all is Permutations and Combinations.
And I want to perfect my ability to do the harder problems in Perms/Combs

I find that for the harder problems, I usually end up not seeing something, end up double counting or not taking into consideration very minor and little things which change the result drastically.

So how do I improve my approach to these problems? Practise doesn't work because I'll just keep doing the problems incorrectly.

Are there maybe any good online resources and stuff to try and approach them properly? i.e. khanacademy etc

Thanks.
 

lochnessmonsta

Booging
Joined
Apr 18, 2012
Messages
157
Gender
Male
HSC
2013
Uni Grad
2017
This is my weakness too. And I understand the 'practice doesnt really help that much' bit, because of the sheer number of possibilities of questions, and that each question looks essentially unique to me. I believe there's a weakness in my understanding of the theory, and not even my teacher can help me.
 

braintic

Well-Known Member
Joined
Jan 20, 2011
Messages
2,137
Gender
Undisclosed
HSC
N/A
I think you just have to keep asking questions here, and see if you can find more connections between apparently different problems.
For example, Ext 2 HSC 1992 Q7a. It is so much easier once you see it is isomorphic to the problem of forming words from the letters RRRRDDDDDD.

Here is a question:

How many solutions consisting of non-negative integers does the equation a+b+c+d=50 have?
(For example a=12, b=13, c=0, d=25 is one solution)

It is actually very easy, but only once you relate it to something more concrete. Apologies if you've already seen this one.

Please could others refrain from jumping in.
 

Sy123

This too shall pass
Joined
Nov 6, 2011
Messages
3,730
Gender
Male
HSC
2013
I refer you to this document: http://www.mediafire.com/?mlxfcg9imjscc4d

Compiled last year for practice.
Thank you.

I think you just have to keep asking questions here, and see if you can find more connections between apparently different problems.
For example, Ext 2 HSC 1992 Q7a. It is so much easier once you see it is isomorphic to the problem of forming words from the letters RRRRDDDDDD.

Here is a question:

How many solutions consisting of non-negative integers does the equation a+b+c+d=50 have?
(For example a=12, b=13, c=0, d=25 is one solution)

It is actually very easy, but only once you relate it to something more concrete. Apologies if you've already seen this one.

Please could others refrain from jumping in.
I see what you mean by simplifying the problem down. For example, if we split 12 people into 3 groups of 4, the ways to do so is:

right? (I think)

But what if 2 people cannot be in the same group? Would it be, first putting those people in the group:



Then selecting people to put in after that:

is this correct?

===

As for your problem, is a good way to simplify it, to consider when none are zero. When a=0, a=b=0 and a=b=c=0?

I can do when a=b=c=0 (obviously) and a=b=0 (26) but I don't know how to a=0 elegantly.
I am thinking of considering the 48 cases for what b could be. And then getting a series to sum up....but I don't think that is what you're looking for.

EDIT: I think I'll watch all of khanacademy's probability playlist, starting from scratch....
 
Last edited:

braintic

Well-Known Member
Joined
Jan 20, 2011
Messages
2,137
Gender
Undisclosed
HSC
N/A
As for your problem, is a good way to simplify it, to consider when none are zero. When a=0, a=b=0 and a=b=c=0?

I can do when a=b=c=0 (obviously) and a=b=0 (26) but I don't know how to a=0 elegantly.
I am thinking of considering the 48 cases for what b could be. And then getting a series to sum up....but I don't think that is what you're looking for.

EDIT: I think I'll watch all of khanacademy's probability playlist, starting from scratch....
Regarding my problem, you don't need to consider cases.
Imagine lining up 50 tokens, and then placing three dividers between them to divide them into 4 groups. Two or more dividers can be adjacent corresponding to zero solutions.
Can you see that all different arrangements of the 50 tokens and 3 dividers correspond to all different solutions to the given equation?
Answer ...?

(Regarding your other question, I'm just taking a break from doing other work, so I'll get back to it later.)
 

Sy123

This too shall pass
Joined
Nov 6, 2011
Messages
3,730
Gender
Male
HSC
2013
Regarding my problem, you don't need to consider cases.
Imagine lining up 50 tokens, and then placing three dividers between them to divide them into 4 groups. Two or more dividers can be adjacent corresponding to zero solutions.
Can you see that all different arrangements of the 50 tokens and 3 dividers correspond to all different solutions to the given equation?
Answer ...?

(Regarding your other question, I'm just taking a break from doing other work, so I'll get back to it later.)
Ah yep I see that, clever, thanks.

right?

Because there are 53 elements to arrange, 3 dividers 50 tokens, and they are not distinct.
 
Last edited:

braintic

Well-Known Member
Joined
Jan 20, 2011
Messages
2,137
Gender
Undisclosed
HSC
N/A
Ah yep I see that, clever, thanks.

right?

Because there are 53 elements to arrange, 3 dividers 50 tokens, and they are not distinct.
That's right. But I prefer to think of it as 53C3. That is, you are picking 3 places from 53 available to place the 3 dividers. Exactly the same thing though.

I think the message is, when you get stuck try to find an isomorphic problem that you can solve.
 
Last edited:

Trebla

Administrator
Administrator
Joined
Feb 16, 2005
Messages
8,401
Gender
Male
HSC
2006
A possibly useful approach for some problems is to look at the most specific example of the specified arrangement and 'multiply out' each dimension of specificity accordingly to get the total count
 

hayabusaboston

Well-Known Member
Joined
Sep 26, 2011
Messages
2,387
Location
Calabi Yau Manifold
Gender
Male
HSC
2013
This is my weakness too. And I understand the 'practice doesnt really help that much' bit, because of the sheer number of possibilities of questions, and that each question looks essentially unique to me. I believe there's a weakness in my understanding of the theory, and not even my teacher can help me.
I c wat u did ther...
 

VBN2470

Well-Known Member
Joined
Mar 13, 2012
Messages
440
Location
Sydney
Gender
Male
HSC
2013
Uni Grad
2017
The whole topic itself can be very difficult as there is such a broad range of questions. However, getting the technique in answering questions is most important when arriving at a solution. I'd say the best way is to expose yourself to as many types of permutations/combinations questions and keep practicing. There is only a set difficulty they can go up to so I'd say practicing lots of questions especially the harder ones really helps. A good book for this is Terry Lee 3U (Fundamental) and the Harder 3U Topics Chapter in the 4U (Advanced) book as well as it covers some very useful techniques. Also past papers helps heaps. Sometimes people get lucky with these types of questions but this topic is never heavily tested in either 3U/4U, only probably a maximum of 2 - 3 marks they can ever ask in the HSC paper (from what I've seen) and sometimes they don't even ask. So don't worry too much about it, everyone is left with uncertainty when answering a question in an exam and perfecting this topic is not as easy as other topics. Practicing is probably the best way to increase your chance to get a question right as well as understanding and setting up a given situation.
 
Last edited:

Makematics

Well-Known Member
Joined
Mar 26, 2013
Messages
1,829
Location
Sydney
Gender
Male
HSC
2013
The whole topic itself can be very difficult as there is such a broad range of questions. However, getting the technique in answering questions is most important when arriving at a solution. I'd say the best way is to expose yourself to as many types of permutations/combinations questions and keep practicing. There is only a set difficulty they can go up to so I'd say practicing lots of questions especially the harder ones really helps. A good book for this is Terry Lee 3U (Fundamental) and the Harder 3U Topics Chapter in the 4U (Advanced) book as well as it covers some very useful techniques. Also past papers helps heaps. Sometimes people get lucky with these types of questions but this topic is never heavily tested in either 3U/4U, only probably a maximum of 2 - 3 marks they can ever ask in the HSC paper (from what I've seen) and sometimes they don't even ask. So don't worry too much about it, everyone is left with uncertainty when answering a question in an exam and perfecting this topic is not as easy as other topics. Practicing is probably the best way to increase your chance to get a question right as well as understanding and setting up a given situation.
2 marks means a lot to some people
 

hit patel

New Member
Joined
Mar 14, 2012
Messages
568
Gender
Male
HSC
2014
Uni Grad
2018
hmm... i am in year 11 and to become better at this all I did was expose myself to a range of question from past papers and books. This is not useless as stated somewhere above in the thread. It helps u recognise the kind of problems and what the examiners are asking for. your impulses become accustomed to that type of question if u do a few. If u dont believe it try it.
 

braintic

Well-Known Member
Joined
Jan 20, 2011
Messages
2,137
Gender
Undisclosed
HSC
N/A
hmm... i am in year 11 and to become better at this all I did was expose myself to a range of question from past papers and books. This is not useless as stated somewhere above in the thread. It helps u recognise the kind of problems and what the examiners are asking for. your impulses become accustomed to that type of question if u do a few. If u dont believe it try it.
I don't think anyone stated that doing lots of questions is useless.

BTW, if you are seriously aiming for 99.95, I think those marks you are aiming for in each subject would see you get about 99.75
 

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top