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|<font color="#FF0000">RedFireTruck</font>]] ([[User talk:RedFireTruck|<font color="#FF0000">talk</font>]]) 12:12, 1 February 2021 (EST) | ||
+ | |||
+ | =Applications= | ||
+ | ==Application 1== | ||
+ | ===Problem=== | ||
+ | ===Solutions=== | ||
+ | ====Solution 1==== | ||
+ | |||
+ | =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.. | ||
+ | |||
+ | "This is a very nice theorem!" - [[User:RedFireTruck|<font color="#FF0000">RedFireTruck</font>]] ([[User talk:RedFireTruck|<font color="#FF0000">talk</font>]]) 10:53, 1 February 2021 (EST) | ||
+ | |||
+ | {{stub}} |
Latest revision as of 17:41, 7 September 2024
Contents
[hide]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 (talk) 12:12, 1 February 2021 (EST)
Applications
Application 1
Problem
Solutions
Solution 1
Testimonials
"Thanks for rediscovering our theorem RedFireTruck" - George and Harry of The Ooga Booga Tribe of The Caveman Society
"This is GREAT!!!" ~ hi..
"This is a very nice theorem!" - RedFireTruck (talk) 10:53, 1 February 2021 (EST)
This article is a stub. Help us out by expanding it.