$n$ distinct objects have $n!$ (linear) permutations and thus $(n-1)!$ circular permutations.
Now consider $m$ objects, some identical, $r_1$ of the first kind, $r_2$ of the second kind, ..., $r_k$ of the $k$th kind. These $n$ objects have $\frac{m!}{r_1!r_2!\dots r_k!}$ (linear) permutations.
Can we likewise reason that these $m$ objects have $\frac{(m-1)!}{r_1!r_2!\dots r_k!}$ circular permutations? I think the answer is no, but can someone explain the intuition why the reasoning that worked earlier doesn't work here? Also, what is the correct number of circular permutations for these $m!$ objects?
(I am hoping for an answer that's suitable for high school students. Thanks.)
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