Doubts in Circular permutation.

Dear Dale,
I'm happy to help with this.

In general, the number of ways to permutate (i.e. arrange the order) of n things is n! See this blog:
https://magoosh.com/gmat/2012/gmat-permu ... binations/

For the folks sitting in a circle, it's tricky because the circle doesn't have a beginning or an end, so we arbitrary pick one element and fix that one person, then arrange everyone else (the other n - 1) in sequence from that one fixed pole -- (n-1)!. This assumes, of course, we are concerned only with arrangements vis-a-vis other people at the table, and we are not concerned, say, with how folks are situated with respect to the room in which we find the table (i.e. who can see out of what window, who faces the door, etc.)

You treated the garland as another circle problem, and that's a very sophisticated way to think about it, but I think the question writer was thinking of a garland having a clear break, a clasp or something where it joins, and that this constitutes an asymmetry, such that we couldn't just rotate the same arrangement of flowers around the garland they way that we could rotate the people around the round table. When we extend the string for the garland in a straight line, as we would when we put flowers on it, then it is not a circle --- it has a clear beginning & end. I suspect the question writer was thinking of it in this form.

I agree, your interpretation is not only valid, but actually mathematically insightful, and because of that, I think there's a problem with the question. Rest assured, my friend, the problem does not lie in your mathematical analysis, but in the semantics of question formulation.

Does this make sense?
Mike