In how many ways can 5 keys be arranged in a | vedclass

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(A) The arrangement of keys in a ring is a case of circular permutation where clockwise and anti-clockwise arrangements are considered identical.The f

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$\frac{1}{2} 	imes 4!$

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(A) The arrangement of keys in a ring is a case of circular permutation where clockwise and anti-clockwise arrangements are considered identical.The formula for circular permutation of $n$ distinct objects is $(n - 1)!$.For a ring or necklace,the number of ways is given by $\frac{(n - 1)!}{2}$.Here,$n = 5$.Therefore,the number of ways $= \frac{(5 - 1)!}{2} = \frac{4!}{2} = \frac{24}{2} = 12$.

In how many ways can $5$ keys be arranged in a ring?

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