View source for Cooga Georgeooga-Harryooga Theorem

=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)