Arrangement Restriction Theorem

The Arrangement Restriction Theorem is discovered by aops-g5-gethsemanea2 and is NOT an alternative to the Georgeooga-Harryooga Theorem because in this theorem the only situation that is not allowed is that all objects are together.

Definition

If there are objects to be arranged and of them should not be beside each other altogether, then the number of ways to arrange them is .

Proof/Derivation

If there are no restrictions, then we have . But, if we put objects beside each other, we have because we can count the objects as one object and just rearrange them.

So, by complementary counting, we get .

Testimonials

I like this theorem, but not as much as the Georgeooga-Harryooga Theorem or the Wooga Looga Theorem ~ ilp

"Very nice theorem but not as impressive as the Georgeooga-Harryooga Theorem." - RedFireTruck