Question

[nsguy1350]

I am sure this is a very easy question, but I am not getting the correct answer when I do this.

Roughly paraphrased: I have three identical pennies and three identical dimes. I want to place them in a circular formation. How many ways can I place the coins if rotation doesn't matter?

Answer:4

I keep getting 20/6, but I am overlooking something very obvious here...

[rcoker7]

On a problem like this, I would recommend just bashing with casework. Using more advanced combinatorics methods just make it overly complicated, when writing out each possibility is much faster(especially since the answer is only 4). I'm curious though, how did you arrive at the answer 20/6?

[nsguy1350]

Well, considering it's 6 different items and it deals with circular permutations, it would be strange and inefficient for me to bash it out. Maybe it would work, but I don't think it would be very good, at least for me.

I did [unparseable or potentially dangerous latex formula] ways to pick three spots, then because it's a circle, divided by 6 to account for the six ways you can rotate it. I remember solving these types of problems before but have forgotten how to do them.

[rcoker7]

The only problem with that method is that not all the cases are counted 6 times. For example, consider the case where the pennies and dimes alternate. Initially(before dividing by 6), you only count this twice(because of the symmetry), so dividing by 6 only counts it as 1/3 of a case in your answer. Adding the missing 2/3 gives you 24/6=4

[nsguy1350]

"Solution 1: You can arrange the coins in 4 ways. DDDPPP , DDPDPP , PPDPDD , and DPDPDP.

The problem is somewhat unclear. Imagine it like this: You are loading your grandma's 6-shooter. You have a 3 red bullets and 3 blue bullets. Grandma is particular about how they are arranged in the gun, but she won't like kill you if she doesn't have to unload. So, how many different ways can you load the gun?

Solution 2:

I define a "circle" of coins to be an arrangement of 6 coins in the 6 fixed spots. This distinguishes between different rotations of an arrangement.

I define a "position" to be an arrangement of 6 coins in 6 unfixed spots. This does not distinguish between rotations of an arrangement.

The only time you can rotate a circle of coins less than 6 places and get the same circle is when it is periodic: PDPDPD or DPDPDP. There are 20 ways to arrange the 6 coins in a line, and thus to arrange them in the 6 fixed spots around the circle. Two of these 20 ways will be the same position: DPDPDP, PDPDPD. All the others come in groups of 6, because you have to rotate them 6 times before it looks the same again. That means there is the one plus (20-2)/3 positions, which is 4."

This is quoted from AuSmith from another thread after a google search on the problem. Apparently, it is a bit different than similar problems I did before. It's a bit tricky, but apparently it didn't trip up Mark Zhang, who got a 360 on this test in competition.

[nsguy1350]

[101dalmatians]

ok now nsguy has qs in math? dont sart in ns :P

[nsguy1350]

Another question: Region 2010, #17. Answer is E (3), I got C (5).

I think what is messing me up is the transition from figure 2 to 3. What does it mean by translate it horizontally? I interpreted it as left squares go right, and right squares return to the left (as in a cycle-type thing. Thus under this definition two translations horizontally produces the same square.) This is the same as right squares go left and left squares go right, so it does not matter which you choose. I know this interpretation must be wrong. I did this sort of problem before, and got it wrong before as well.

Also, it is frustrating when I get the correct answer but put the wrong answer on my answer sheet (did this at TMSCA State... cost me places). I did this twice on this test, except I got the right answer but circled the wrong one (which cost me places in the state-wide rankings of that year.) Does anyone else do this too?

[darksaber21]

It's a literal translation, i.e the spots are basically in the same position relative to each other.


When they say translation like that, they mean literal movement on the page, like just moving the spots as if it was there in real life. Does that make any sense?




And that happens to me a lot, especially when I'm reckless.

[nsguy1350]

[darksaber21]

[101dalmatians]

[sxk1693]

[101dalmatians]