Hat-Check Problem/Examples/3
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Example of Hat-Check Problem
- $p_3 = \dfrac 1 3$
Proof
When $n = 3$, there are only three hats to hand back and $6 = 3!$ ways to return the hats to the owners.
- $\begin{array}{c|ccc}
\ds Correctly Returned & Adam & Bill & Clyde \\ \hline
3 & A & B & C \\ \hline
1 & A & C & B \\ \hline
1 & C & B & A \\ \hline
1 & B & A & C \\ \hline
0 & B & C & A \\ \hline
0 & C & A & B \\ \hline
\end{array}$
From the table above, we see that there are only two permutations that irritate all parties involved.
Hence, $p_3 = \dfrac 2 6 = \dfrac 1 3$
$\blacksquare$