Difference between revisions of "Cooga Georgeooga-Harryooga Theorem"
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Now we have <math>a-b</math> blanks and <math>b</math> other objects so we have <math>_{a-b}P_{b}=\frac{(a-b)!}{(a-2b)!}</math> ways to arrange the objects we can't put together. | Now we have <math>a-b</math> blanks and <math>b</math> other objects so we have <math>_{a-b}P_{b}=\frac{(a-b)!}{(a-2b)!}</math> ways to arrange the objects we can't put together. | ||
| − | By | + | By The Fundamental Counting Principal our answer is <math>\frac{(a-b)!(a-b)!}{(a-2b)!}</math>. |
Revision as of 19:42, 31 January 2021
Contents
Definition
The Cooga Georgeooga-Harryooga Theorem (Circular Georgeooga-Harryooga Theorem) states that if you have 
distinguishable objects and 
objects are kept away from each other, then there are 
ways to arrange the objects in a circle.
Created by George and Harry of The Ooga Booga Tribe of The Caveman Society
Proofs
Proof 1
Let our group of objects be represented like so 
, 
, 
, ..., 
, . Let the last objects be the ones we can't have together.
Then we can organize our objects like so ![[asy] label("$1$", dir(90)); label("BLANK", dir(60)); label("$2$", dir(30)); label("BLANK", dir(0)); label("$3$", dir(-30)); label("BLANK", dir(-60)); label("$\dots$", dir(-90)); label("BLANK", dir(-120)); label("$a-b-1$", dir(-150)); label("BLANK", dir(-180)); label("$a-b$", dir(-210)); label("BLANK", dir(-240)); [/asy]](http://latex.artofproblemsolving.com/b/d/e/bded4c8eb437eac7c96ffdda3c7f36dc774fb40c.png)
We have 
ways to arrange the objects in that list.
Now we have 
blanks and other objects so we have 
ways to arrange the objects we can't put together.
By The Fundamental Counting Principal our answer is 
.
Proof by RedFireTruck
Testimonials
"Thanks for rediscovering our theorem RedFireTruck" - George and Harry of The Ooga Booga Tribe of The Caveman Society
"This is GREAT!!!" ~ hi..
