Given finite set $X$ with $n$ elements, how many permutations are there of $X$?

1. Show that if $X=\{1\}$, then there is only one permutation of $X$.
2. Show that if $X=\{1,2\}$, then there are two permutations of $X$. List them.
3. Make a list of all permutations of $X=\{1,2,3\}$, and then of $X=\{1,2,3,4\}$. What pattern do you see?
4. Make a conjecture about the number of permutation of $X$ if $|X|=n$. Then prove your conjecture.

• Make remarks with a list.

problem.number_of_permutations.tex

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\begin{problem}
Given finite set $X$ with $n$ elements, how many permutations are there of $X$?
\begin{enumerate}
Show that if $X=\{1\}$, then there is only one permutation of $X$.
\item Show that if $X=\{1,2\}$, then there are two permutations of $X$. List them.
\item Make a list of all permutations of $X=\{1,2,3\}$, and then of $X=\{1,2,3,4\}$. What pattern do you see?
\item Make a conjecture about the number of permutation of $X$ if $|X|=n$. Then prove your conjecture.
\end{problem}

• Dihedral group

problem ben