Difference between revisions of "Cooga Georgeooga-Harryooga Theorem"
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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. | 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 < | + | 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. | We have <math>(a-b)!</math> ways to arrange the objects in that list. |
Revision as of 18:33, 31 January 2021
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 fundamental counting principal our answer is .
Proof by RedFireTruck