HSC q16 a ii (1 Viewer)

[swagswagyoloswag]

*HSC 2012
How would I do this?

(ii) Consider three identical separators to divide
the 10 coins into 4 boxes, total = 13 items, including
3 identical separators and 10 identical coins.
This was from Terry Lee's solutions, he referred to his book but I don't have it.
What's a "divider" and can someone explain his solution? as well as any other alternate solutions

 

[Drongoski]

This is a common technique for such a problem.

Imagine you have a word 'aajaaajajaaaa' where the 'a's represent the coins and the 'j's represent the separators. You have learnt that if a word has 10 'a's and 3 'j's, you can form different 'words', including the one cited, which represents the case of 2 coins in box 1, 3 coins in box 2, 1 coin in box 3 and 4 coins in box 4. etc etc etc



Well Braintic has already done it!

 

[braintic]

This is about counting the number of ways of distributing 10 identical coins amongst 4 distinguishable boxes.

Lay the 10 coins in a line:

OOOOOOOOOO

Divide them into 4 groupings by placing 3 dividers between them.

For example:

OOO|OOOO|O|OO
means place 3 coins in box A, 4 in box B, 1 in box C, 2 in box D

||OOOOOOOOO|O
means place 0 in box A, 0 in box B, 9 in box C and 1 in box D

Each unique "word" that can be formed from 10 O's and 3 |'s represents a unique distribution of coins.

The number of such words is 13! / (9! 4!)

 

[swagswagyoloswag]

Thank you. Is there some place that shows us all the possible techniques used in this topic? Ive only heard about this right now
and are there any specific exercises that have these types of problems?

 

[braintic]

Thank you. Is there some place that shows us all the possible techniques used in this topic? Ive only heard about this right now
and are there any specific exercises that have these types of problems?

I'm not aware of such a place. The idea is to convert the problem into a problem that has the same solution, but which is easier to visualise. There are a multitude of possible scenarios.

Has your teacher finished teaching the topic?

 

[InteGrand]

Thank you. Is there some place that shows us all the possible techniques used in this topic? Ive only heard about this right now
and are there any specific exercises that have these types of problems?

This method is called Stars and Bars: https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)

You can probably search for stars and bars combinatorics problems online.

 

[swagswagyoloswag]

I'm not aware of such a place. The idea is to convert the problem into a problem that has the same solution, but which is easier to visualise. There are a multitude of possible scenarios.

Has your teacher finished teaching the topic?

No, but she just does questions from the textbook, doesn't show us many concepts and ways to do it

 

[Drongoski]

This method is called Stars and Bars: https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)

You can probably search for stars and bars combinatorics problems online.

The stars & bars technique was used and introduced by William Feller, in his textbook "An Introduction to Probability Theory and Its Applications" (John Wiley) - first published in 1950 (I happen to have Vol 1 (3rd Edn)of the book).

 

[Drsoccerball]

What if theres a restriction to the amount on each finger/box or whatever?

 

[braintic]

What if theres a restriction to the amount on each finger/box or whatever?

Minimum or maximum?