Circular Combinatorics Question help! (1 Viewer)

[Epicman69]

How do you arrange something in a circular bracelet with 8 beads where 3 of the beads are the same colour?

 

[idkkdi]

How do you arrange something in a circular bracelet with 8 beads where 3 of the beads are the same colour?

nvm i think I understood it wrong lol

 

[ExtremelyBoredUser]

Would it be 60 since you could flip the bracelet since its considered a bead property? I'm not entirely sure but I remember stumbling across beads type questions and you would solve them normally as a circle arrangement question and then divide it by 2.

5!/2 = 120/2 = 60

The three beads are the same color so you couldn't arrange them inside the group and arranging other beads you would get 5! and then divide by 2? I'm probably wrong but these questions confuse me a bit as well.

I really wish I could show a diagram, I'll try to draw my point of view if I can.

 

[Epicman69]

nvm I just got the answer its 7!/3!

 

[ExtremelyBoredUser]

nvm I just got the answer its 7!/3!

oh nvm I guess this question was considered a normal circle arrangements problem. 7! for normal arrangement in a circle and you divide it by 3! because there's 3 identical beads, just like how you would do with arrangements with more than 1 same letter.

 

[Epicman69]

What did you do?

It's just like putting 8 beads into a circle right? So you get 7! but you need to divide by 3! since there are 3 beads of the same colour and as such there is 3! repetitions of the same pattern, note that the question never mentioned that you need the beads to be together.

 

[idkkdi]

nvm I just got the answer its 7!/3!

that's what I thought at first, but I don't think it works like that.

 

[Epicman69]

that's what I thought at first, but I don't think it works like that.

idk but that was the right answer so...

 

[ExtremelyBoredUser]

that's what I thought at first, but I don't think it works like that.

Yeah normally with questions that refer to beads are one's you divide by two but I'm unsure that's right as well

 

[ExtremelyBoredUser]

idk but that was the right answer so...

Yeah this is what annoys me about the topic, you do a truckload of questions with that style and you use one strategy to do them and you get them right, but then they throw another question but suddenly it has different properties.

Like I swear questions with beads you have to divide by two because they produce same arrangements when you "flip it" or so I've been taught that way, but in this one you just do it like a regular question which would ask you to arrange, for say MATHEMATICS, in a circle and you would do (n-1)!/2!2!2! etc.

 

[Epicman69]

Yeah this is what annoys me about the topic, you do a truckload of questions with that style and you use one strategy to do them and you get them right, but then they throw another question but suddenly it has different properties.

Like I swear questions with beads you have to divide by two because they produce same arrangements when you "flip it" or so I've been taught that way, but in this one you just do it like a regular question which would ask you to arrange, for say MATHEMATICS, in a circle and you would do n!/2!2!2! etc.

yea ikr, I hate how you can't really check if you got it right.

 

[ExtremelyBoredUser]

yea ikr, I hate how you can't really check if you got it right.

I guess you just have to go with what methods your teacher taught you for the topic because its most likely the correct one for examination purposes. Really annoys me when there's two contradicting methods from different textbooks or sources which produce different answers; at least with other topics there's only a clear cut answer such as polynomials or trig, you either screwed up in your working out or you just flat out don't know how to do it. I don't hate this topic but when it pulls shit like this...

 

[idkkdi]

How do you arrange something in a circular bracelet with 8 beads where 3 of the beads are the same colour?

are the other 5 beads the same colour or different.

 

[Epicman69]

are the other 5 beads the same colour or different.

since they didn't mention anything about them I presumed that you assume the other beads are different

 

[idkkdi]

since they didn't mention anything about them I presumed that you assume the other beads are different

if the answer was 7C3, the other 5 are also the same colour.

Now for @ExtremelyBoredUser's point about flipping. I'm pretty sure that is meant to be accounted for and that the book forgot or smth.

Answer should be 8C3 /(8*2)
8 for rotation.
2 for flipping.

 

[Life'sHard]

I did 7!/(3!*2!). 7! for the arrangement of the rest of the beads. Divide by 3! since identical and 2! since it's a bracelet.

 

[ExtremelyBoredUser]

I believe the book just considered beads as a placeholder like people, objects, letters instead of considering the effects of properties so the answer was 7!/3! as a result since they were probably considering it a circle arrangement.

If the beads property was considered, then the answer would've been what @Life'sHard did.

And here's additional context to the whole flipping thing I'm talking about;


Source:

Mathematics Extension 1 - Permutations and Combinations - Dux College

Permutations and Combinations DefinitionsContents1 Definitions2 The Fundamental Counting Principle2.1 Example 12.2 Solution 12.3 Example 22.4 Solution 23 Permutations and the …

dc.edu.au

I didn't learn it off here but it explains the "flipping" thing I'm referring to. The beads questions in the cambridge textbook used this same process and so did other test papers/sources of questions, so I assumed the same here; the question really didn't give any indicators to assume or not to assume so

 

[Epicman69]

I believe the book just considered beads as a placeholder like people, objects, letters instead of considering the effects of properties so the answer was 7!/3! as a result since they were probably considering it a circle arrangement.

If the beads property was considered, then the answer would've been what @Life'sHard did.

And here's additional context to the whole flipping thing I'm talking about;

View attachment 31896
Source:

Mathematics Extension 1 - Permutations and Combinations - Dux College

Permutations and Combinations DefinitionsContents1 Definitions2 The Fundamental Counting Principle2.1 Example 12.2 Solution 12.3 Example 22.4 Solution 23 Permutations and the …

dc.edu.au

I didn't learn it off here but it explains the "flipping" thing I'm referring to. The beads questions in the cambridge textbook used this same process and so did other test papers/sources of questions, so I assumed the same here; the question really didn't give any indicators to assume or not to assume so

I think I should go with this since this is probably the correct way to do the question anyways cause it considers more variables i.e flipping the bracelet.

 

[azizcuber]

funny number

I think I should go with this since this is probably the correct way to do the question anyways cause it considers more variables i.e flipping the bracelet.