## Friday, September 28, 2007

### Another Counting Problem

In how many different ways can 6 objects be arranged in a circular pattern?

1. Well, let's look at how many ways they can be arranged in a linear pattern. This is a classic factorial problem, so the answer to that is 6!

However, when you make the pattern circular, for each pattern that you have, the "starting number" is arbitrary. This means that 123456 = 234561 = 345612 = ...

So, each pattern with "1" as the starting number takes care of 5 others (1 pattern -> 6 patterns). So, I'd just divide by 6.

6!/6 = 5*4*3*2*1 = 120

2. I think you've figure it out Abe. It actually wasn't the answer I was thinking of, but your answer makes a lot of sense.

3. Circular permutations. From a placement point of view, ask the first person to sit. Doesn't matter where. Now, the second person has to make a choice, and the third, etc.
1 x 5 x 4 x 3 x 2 x 1

4. Ofcourse the answer is gonna be 120 since we would have to find the answer by keeping one object fixed and then finding the possible ways.

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