Answer:
[tex]840[/tex]
Step-by-step explanation:
This is a good example of permutations. For the first slot on his shelf, Dylan has 7 trophies to choose from. Then 6, 5, and so on. Since he is only fitting 4 trophies on his shelf, we only continue until we have the slots filled. Thus, we have [tex]7\cdot 6 \cdot 5\cdot 4=\boxed{840\:\text{ways}}[/tex] to do so.
*Note: We do not divide by [tex]4![/tex] because in this case, order matters. A different order of the same four books still produces different arrangements, so the order matters.
