Exactly 4 actors try out for the 4 parts in a play. if each actor can perform any one part and no one will perform more than one part, how many different assignments of actors are possible?

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You have 4 choices for the first actor, 3 choices for the secnod, 2 choices for the third, and 1 choice for the fourth.  Therefore, you have 4*3*2*1=24.  This can also be written as 4 factorial, or 4! = 24.