$In how many different ways $n$ persons can seat around a round table?$

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$In how many different ways $n$ persons can seat around a round table?$

I am confused about counting all possible ways that $n$ persons can seat around a round table.

Discrete Mathematics Combinatorics Permutation

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Let the first person seat. Since the table is round, all $n$ options of the first person are similar. Hence the first person can seat in only one different way. After the first person seats the second person has $n-1$ different seats to choose from. The third person would have $n-2$ options and so on. Hence the number of different ways that $n$ persons can seat around a round table is
\[1 \times (n-1) \times (n-2) \times \cdots 2\times 1=(n-1)!.\]

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$In how many different ways $n$ persons can seat around a round table?$

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