Difference between revisions of "Cooga Georgeooga-Harryooga Theorem"
Redfiretruck (talk | contribs) |
Redfiretruck (talk | contribs) |
||
Line 9: | Line 9: | ||
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 19:33, 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
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