ANSWER
840
EXPLANATION
Lauren has 7 trophies and wants to know how many possible arrangements with 4 trophies there are. If we want to know how many different arrangements, then order matters so we would use a permutation.
But we can also think it logically and we will get the same answer if we do it right. Let's see, for the first shelf space, there are 7 trophies available to pick from. Then for the second shelf space, there are 6 trophies left. Then, for the third space, there are 5 trophies left and, finally, for the fourth space there are 4 trophies to pick from,
[tex]7\cdot6\cdot5\cdot4=840[/tex]
If we do it with a permutation,
[tex]_7P_4=\frac{7!}{(7-4)!}=\frac{7!}{3!}=\frac{7\cdot6\cdot5\cdot4\cdot3!}{3!}=7\cdot6\cdot5\cdot4=840[/tex]
Hence, there are 840 possible arrangements.
