Difference between revisions of "Cooga Georgeooga-Harryooga Theorem"
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− | + | =Definition= | |
+ | The Cooga Georgeooga-Harryooga Theorem (Circular Georgeooga-Harryooga Theorem) states that if you have <math>a</math> distinguishable objects and <math>b</math> objects are kept away from each other, then there are <math>\frac{(a-b)!^2}{(a-2b)!}</math> ways to arrange the objects in a circle. | ||
+ | |||
+ | |||
+ | Created by George and Harry of [https://www.youtube.com/channel/UC50E9TuLIMWbOPUX45xZPaQ The Ooga Booga Tribe of The Caveman Society] | ||
+ | |||
+ | =Proofs= | ||
+ | ==Proof 1== | ||
+ | Let our group of <math>a</math> objects be represented like so <math>1</math>, <math>2</math>, <math>3</math>, ..., <math>a-1</math>, <math>a</math>. Let the last <math>b</math> 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> | ||
+ | |||
+ | We have <math>(a-b)!</math> ways to arrange the objects in that list. | ||
+ | |||
+ | 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 The Fundamental Counting Principal our answer is <math>\frac{(a-b)!^2}{(a-2b)!}</math>. | ||
+ | |||
+ | |||
+ | Proof by [[User:RedFireTruck|RedFireTruck]] | ||
+ | |||
+ | =Testimonials= | ||
+ | "Thanks for rediscovering our theorem [[User:Redfiretruck|RedFireTruck]]" - George and Harry of [https://www.youtube.com/channel/UC50E9TuLIMWbOPUX45xZPaQ The Ooga Booga Tribe of The Caveman Society] | ||
+ | |||
+ | "This is GREAT!!!" ~ hi.. |
Revision as of 10:37, 1 February 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
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..