# 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 .

## Problem

Alice, Bob, Carl, David, Eric, Fred, George, and Harry want to stand in a line to buy ice cream. Fred and George are identical twins, so they are indistinguishable. Alice, Bob, and Carl **cannot be altogether** in the line.

With these conditions, how many different ways can you arrange these kids in a line?

Problem by Math4Life2020, edited by aops-g5-gethsemanea2

### Solution

By the Arrangement Restriction Theorem, we get because Fred and George are indistinguishable.

Solution by aops-g5-gethsemanea2

## 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