A certain football coach allows his quarterback to call three plays in a row. If the quarterback can choose from 3 passing plays, 4 running plays, and 2 trick plays, how many different arrangements of plays can be run, if the same play can’t be run more than once?  Choose the correct choice from the following options: A) 24 B) 125 C) 243 D) 504 E) 729
P=3, R=4, and T=2.   There are following cases possible:   1. Different kinf of plays:   PRT (Any combination) = 3*4*2*3!=144.   2. Two same kind of plays and one from other kind of play:   PPR (Any combination. Notice that the two P's here must be different) = C23∗4∗3!=72C32∗4∗3!=72;   PPT = C23∗2∗3!=36;   RRP = C24∗3∗3!=108   RRT = C24∗2∗3!=72   TTP = C22∗3∗3!=18   TTR = C22∗4∗3!=24   3. All three plays are of the same kind:   PPP = C^3_3*3!=6   RRR = C^3_4*3!=24   TTT is not possible because we have only two T's.   Total = 144 + 72 + 36 + 108 + 72 + 18 + 24 + 6 + 24= 
<gadget id="calculator">144 + 72 + 36 + 108 + 72 + 18 + 24 + 6 + 24</gadget>
<output>504</output>
504.   Answer: 504.
<result>D</result>
A certain football coach allows his quarterback to call three plays in a row. If the quarterback can choose from 3 passing plays, 4 running plays, and 2 trick plays, how many different arrangements of plays can be run, if the same play can’t be run more than once?