Help/Assistance: Permutations + Combinations (1 Viewer)

[puff.romeo]

hey

um, this is my first post. so um.. not quite sure if im doing this right. but anyway.. i need some help with my prelim permutations + combinations revision im doing right now.

i've got a test coming up and i havent done permutations and combinations in class cos i was on a holiday. anyway. my question is:

according to my preliminary extension 1 excel maths book,
p.g. 65
"In how many ways can seven different beads be placed in a circle to form a necklace?"
the answer is
the number of ways = (6!)/2 = 360
because in this case both positions give the same necklace.

"if no distinction is made between clockwise and anticlockwise arrangements, the number of arrangements of n distinct objects in a circle is
[(n-1)!]/2


ok. so i get that part.

now why is it that in another question for e.g.

in how many ways can the numbers 1,2,3,4,5,6 be arranged around a circle
equals to
(6-1)! = 5! = 120

what's the difference. why is that the first one u had to divide by two.. and int he second question u didnt?
from what i can understand.. the second question "didnt make a distinction between clockwise and anticlockwise arrangements either"

that's what i don't understand.. in what sort of circumstances do i divide by two.. all question that mention beads lol? or...


thanks in advance. lots.

-puff.romeo

 

[Riviet]

Say you're holding a necklace in your hand and you place it on a table. There are (n-1)! different combinations of those beads. Now if you flip it over, each of these combinations will exactly be the same. That's why you divide by two for questions where clockwise and anti-clockwise are the same.
However, if you have a question involving people sitting around a circular table, there are (n-1)! ways, however, you can't just flip the table over for any anti-clockwise combinations, that's why you don't divide by two. I hope that helps.

 

[puff.romeo]

oh. ok.
i think i understand. but when you are calculating the total amount of possible combinations, you don't flip the necklace over and count those as well do you? because the way i see it, we shouldn't have to divide by two if you aren't counting those anyway. right?

im really confused. but thanks for the help. i'm going to try and get my head around it.

another question.. in what sort of circumstances should i divide by two?

i.e. how do i tell if a question implies that i should divide by two.. what are the differences between the questions..

thanks~

puff.romeo

 

[Riviet]

another question.. in what sort of circumstances should i divide by two?
i.e. how do i tell if a question implies that i should divide by two.. what are the differences between the questions..

You need to divide by two when anti-clockwise and clockwise arrangements are considered the same. Examples of when you need to divide by two are necklaces, bracelets, anything that you can easily flip over, maybe a tambourine, can't think of anymore.
Sometimes it will be obvious if the question specifies if anti-clockwise/clockwise are different or the same thing, but most of the time, you have to figure it out for yourself.

 

[puff.romeo]

oh alright then. gotcha.


thanks a tonne riviet.

 

[mightybeav]

Ok, i need some help also. Here is the question:

Driving with my son the other day, i hand him a package of necco candies. He says to me, there is no other package like this in the world. I said that is not true.

Now there are 39 pieces and 8 colors. how many unique packages are there?

please show me your calculation, it has been a while since i have done this.

Thanks

Mighty

 

[mightybeav]

wow, thanks. does that take into account that they could all be one color?