A querie about order?

[peter]

Hope someone with a bit of maths nous can help me out with this.

Google tells me that if you have 6 books to place on a shelf there are 720 ways in which they can be ordered. Say we call them a to f, we could start with say a,b,c,d,e,f. Then we could begin the rearrangement by moving along the line sequentially b,a,c,d,e,f......b,c,a,d,e,f.....b,c,d,a,e,f......b,c,d,e,a,f and finally b,c,d,e,f,a.

We might then move on to moving b along the line - so we go a,c,b,d,e,f....etc. If we had started with b at the beginning of the line in this second (b part) of the above plan, we would have replicated the b,a,c,d,e,f of the first section and our final list (sorry - this is what we are after) would be wrong. Unless the scheme we follow in constructing our list is one that precludes such repetitions our final list will contain many such, and will be way over the 720 we require.

Can anyone help me by providing such a scheme for me to follow that results in a complete list, but makes such repetitions either not occur - or is so idiot proof that they are easy to spot?

Gratitude for any help that can be provided!

[Vraith]

unless I"m quite confused about what you're asking, 720 IS the unique/non-repetitive number. You do that as factorials [6x5x4x3x2x1=720].
The set including repetitions would be elements to the elements power. [[in this case 6 to the 6th].

[samrw3]

Peter I am just guessing on what might be causing your confusion so here goes nothing.

In factorial math the calculation assumes no possibilities of repetition. The calculation purpose is to discover how many unique ways of six books can those six books be arranged. Thus since it is unique ways there is the assumption that no repetition is occurring.

My guess is you are continually swapping out letters in a row which could allow for repetition. Although that is one possible way to organize the books - allowing for repetition - you no longer have unique ways and thus this cannot be solved by normal factorial math.

[peter]

Ok - I'm not making myself clear; I'm doing a picture with six different elements that will be arranged linearly according to the 720 different orders that six things can be arranged in. It's important that I don't screw up by either missing out or repeating orders. So I need a scheme such as the one I began above, that will give me a means of writing down the 720 different orderings without messing up - because a) I'm not smart enough to work out how to do it myself and b) I'm not sharp enough to be able to look at shed loads of different orderings and spot if I've replicated any (and neither I'm thinking is anybody else who hasn't got an autistic/Asperger's streak in them.

Any ideas?

[peter]

I've found something that might help;

Somebody asked how many ways the six letters of the word Sunday could be arranged (720 as we know) and was given the following answer......

The first letter of the rearrangement can be any one of six letters.

The second letter can now be any one of five letters, so the first two letters can be a combination of 6 times 5 letters or 30 arrangements

The third letter can be any one of four remaining letters, so the first three letters can be a combination of 6 times 5 times 4 letters or 120 arrangements

The fourth letter can be any one of three remaining letters, so the first four letters can be a combination of 6 times 5 times 4 times 3 letters or 360 arrangements

The fifth letter can now be only one of two remaining letters, so the first five letters can be a combination of 6 times 5 times 4 times 3 times 2 letters or 720 arrangements

The final letter can now be only one letter because all the others have been used. So the six letters can be a combination of 6 times 5 times 4 times 3 times 2 times 1 letters or 720 arrangements

This at least is helpful in understanding how the 720 figure is arrived at even if it doesn't actually help me to write the 720 different orders down!

[samrw3]

I deleted SUNDAY - but 1-6 is few posts below.

[samrw3]

There are the first 40 rows of SUNDAY[ now deleted since I changed to 1-6]. I could send you all 720 rows but needed to make sure this would be helpful.

Because the largest limitation is that this is a picture and thus cannot be easily manipulated to different symbols. (unless you know more about picture manipulation then I do)

However if you send me the symbols that you want I can try sending you all 720 variations of those symbols. I created a table where creating all 720 variations is easy. The tricky part is grabbing the snapshot(picture) of each section and uploading it to KW. [Thus why I don't want to load multiple sections of pictures if you cannot work with a picture format]

Also I may not be able to create 720 variations if they are strange symbols that I cannot easily (or perhaps not at all) reproduce on a standard keyboard.

[peter]

Sam - you are a genius!

I've been trying until I'm blue in the face to find a list of the 720 variations (or a way to create one) to no avail. Shed loads of sites tell you how to calculate the number of alternatives - even how to calculate the position of a given number (is it's place) in an ascending list if the content is numerical - but not one tells you how to build the list itself!

Ok. Because of an interest in abstract art (well, art generally but abstract in particular at present) I decided to do a painting that featured all the various orders in which 6 colours could be linearly placed.

For practical purposes the numbers 1 to 6 would be perfect, since I am so familiar with the coler/number associations I have made that I am almost synesthetic in respect to them now. A list therefore of all 720 variations featuring the first six integers would be perfect!

Can't promise the finished work will be very good, but I give my word you'll get the chance to judge!

(It's my first abstract work, though I've done some acceptable figurative pieces before - strictly amateur I hasten to add!)

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