On combinatorial arguments (1 Viewer)

[Kurosaki]

Hello BOS denizens .

Do you know whether the HSC allows for combinatorial proofs when dealing with binomial coefficient identities?

e.g. Proving that

We can model this as creating a committee of size k people with a single chairperson, from a total amount of n people.
We can do this by: first choosing k people from n, then from those k people choosing a single person. Summing gives the LHS of the above expression.

Alternatively, choose one person to be the chairperson from the n people, done in n ways. Then for each subsequent person, they can either be in the committee or not, which accounts for the of the RHS. Multiplication principle gives the required expression.

And as these 2 expressions are different ways of counting the same thing, they are equal.

Combinatorial proofs seem more natural to me than the mechanical method of considering a geometric series and comparing coefficients, and in some instances I find a bit faster as well..

Thanks in advance to anyone that helps with this!

 

[Carrotsticks]

You most certainly can, provided that the question does not set out a specific structure for you to do it (ie: "Hence" or "Deduce").

This is why they say "Hence, or otherwise", though very few students take the "Or otherwise" approach.

Be careful though because often, in the excitement of knowing the result of your combinatoric argument, it is possible for you to accidentally fudge something to suit the result, since you know what it's supposed to be!

 

[Kurosaki]

Right, thanks Carrot!

 

[Fade1233]

Hello BOS denizens .

Do you know whether the HSC allows for combinatorial proofs when dealing with binomial coefficient identities?

e.g. Proving that

We can model this as creating a committee of size k people with a single chairperson, from a total amount of n people.
We can do this by: first choosing k people from n, then from those k people choosing a single person. Summing gives the LHS of the above expression.

Alternatively, choose one person to be the chairperson from the n people, done in n ways. Then for each subsequent person, they can either be in the committee or not, which accounts for the of the RHS. Multiplication principle gives the required expression.

And as these 2 expressions are different ways of counting the same thing, they are equal.

Combinatorial proofs seem more natural to me than the mechanical method of considering a geometric series and comparing coefficients, and in some instances I find a bit faster as well..

Thanks in advance to anyone that helps with this!

HSC allows anything, if it works and is right. -Terry Lee

 

[Carrotsticks]

HSC allows anything, if it works and is right. -Terry Lee

Not always true.

For example, 2009 Extension two HSC. Finding the square root of 3+4i.

Students were not awarded full marks for writing down the answer, even if it was right. They specifically stated that in the markers comments.

 

[seventhroot]

Not always true.

For example, 2009 Extension two HSC. Finding the square root of 3+4i.

Students were not awarded full marks for writing down the answer, even if it was right. They specifically stated that in the markers comments.

huehuehue

TL has that special method in his Adv maths book which tbh I never got my head around

but yeah as carrot said; my teacher was very lenient on giving marks out but the HSC may not be that way