Difference between revisions of "Arrangement Restriction Theorem"

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So, by complementary counting, we get <math>n! - (n - k + 1)!k!</math>.
 
So, by complementary counting, we get <math>n! - (n - k + 1)!k!</math>.
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==Testimonials==
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I like this theorem, but not as much as the Georgeooga-Harryooga Theorem or the Wooga Looga Theorem ~ ilp

Revision as of 09:40, 20 December 2020

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 $k$ objects are together.

Definition

If there are $n$ objects to be arranged and $k$ of them should not be beside each other altogether, then the number of ways to arrange them is $n! - (n - k + 1)!k!$.

Proof/Derivation

If there are no restrictions, then we have $n!$. But, if we put $k$ objects beside each other, we have $(n-k+1)!k!$ because we can count the $k$ objects as one object and just rearrange them.

So, by complementary counting, we get $n! - (n - k + 1)!k!$.

Testimonials

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

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