Problem (1.21 from MC II-A Combinatorics) for reference:

First off, note that you do get the same answer each way, so it is okay to use cases, but they are not required here.

The key here is that actually NONE of the choices affect other choices. (This is different from one of the examples involving shorts vs pants). Thus, we are able to make all these choices independently from the others (and can use the Product Rule) to multiply out the number of outcomes).

I like thinking of a problem like this as "ordering an outfit from a menu". Note we can make our menu:

• Shirt: Shirt A, ..., or Shirt D ($4$ choices)
• Legwear: Shorts A, ..., Shorts E, Pants A, ..., Pants C ($8$ choices)
• Shoes: Shoes A, ..., Shoes E ($5$ choices)
• Hat: Hat A, Hat B, or No Hat ($3$ choices)
• Jacket: Jacket A or No Jacket ($2$ choices)

Then, multiplying we get our answer $4\cdot 8\cdot 5\cdot 3\cdot 2 = 768$. Hope this helps!