A key ring has 7 keys. How many different ways can they be arranged?
I would like to argue against OA. 7 => 6! circular permutations. Therefore, for a keyring - arrangements = 6!/2 (clockwise/anti-clockwise doesn't matter)
Think of choosing one specific key and placing it on the key ring. The remaining 6 keys can now be arranged in 6! ways. Once you have placed one key, you created a row (straighten the circle), so the regular rule for permutations apply.
It doesn't matter if you slide the keys clockwise or counter clockwise, they have the same relative position each to the other. For example, just for 3 keys, ABC and ACB are different arrangements. Mirror image arrangements are different, because you don't flip your key ring. At least I think this is what is assumed.
In the case of arrangements around a round table, it is obvious. For sure, you cannot flip your table and the people with it.
PhD in Applied Mathematics
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