Revision as of 18:34, 31 January 2021

Definition

The Cooga Georgeooga-Harryooga Theorem (Circular Georgeooga-Harryooga Theorem) states that if you have $a$ distinguishable objects and $b$ objects are kept away from each other, then there are $\frac{(a-b)!(a-b)!}{(a-2b)!}$ 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 $a$ objects be represented like so $1$ , $2$ , $3$ , ..., $a-1$ , $a$ . Let the last $b$ objects be the ones we can't have together.

Then we can organize our objects like so

label("$1$", dir(90));
label("_", dir(60));
label("$2$", dir(30));
label("_", dir(0));
label("$3$", dir(-30));
label("_", dir(-60));
label("$\dots$", dir(-90));
label("_", dir(-120));
label("$a-b-1$", dir(-150));
label("_", dir(-180));
label("$a-b$", dir(-210));
label("_", dir(-240));
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We have $(a-b)!$ ways to arrange the objects in that list.