For a special opening act in a talent show, \(8\) performers (\(3\) singers, \(3\) dancers and \(2\) musicians) are to stand in a circle on stage. Find the number of possible arrangements if
there are no restrictions,[1]
all performers of the same role must stand together.[2]
For the main performance, \(12\) performers (\(3\) singers, \(5\) dancers and \(4\) musicians) are arranged in a line on stage such that no two musicians are adjacent. Find the number of possible arrangements.[2]
\(7\) performers are selected from \(15\) performers (\(6\) singers, \(5\) dancers and \(4\) musicians) for the closing act. Find the number of possible selections if there must be at least \(2\) performers from each role.[3]
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Answer:(a)(i) \(5040\). (ii) \(144\). (b) \(121\,927\,680\). (c) \(2700\).