What size wire is needed from a 200 amp home box going 175 feet to a panel in detached garage. Would it be 3 wire or 4 wire?
In the current era, for a garage you had better plan a feed that can support EV charging, at least, even if you don't currently plan to do that. You've provided no load information, so picking a wire size is speculative at best.
4 wire is required, barring some exceptions that are unlikely to come into play economically at this distance (such as using transformers for longer distances, but they would not be cost-effective at this distance .vs. larger wire.)
Use conduit (trenches are expensive, conduit is cheap) and put in at least one spare conduit for communications circuits, since (trenches are expensive, conduit is cheap.)
Use a large panel with many spaces - the amp rating of the panel is mostly irrelevant (you can have a 200A main breaker panel and feed it with a 50A breaker - the 200A main is just serving as the required switch in that case.)
When people ask "3-wire or 4-wire", that question usually means "Do I need to run a ground wire, or not?"
The answer is Yes / 4-wire -- You need to run a ground wire. You may also need ground rods, but one is not a substitute for the other.
We love saving money here. We have tricks that'll save you 65% on wire for instance. However one place we NEVER try to chintz out is the subpanel. We always recommend a super large subpanel, with many, may spaces -- because spaces are dirt cheap, running out of spaces is an expensive, frustrating and unnecessary problem. So go big - and that almost always means using a subpanel whose main breaker or busing (e.g. 125A) is larger than your feeder. That is fine. That's a "never exceed" safety limit - like 112 mph tires or 150 psi compressor tank. Better not to even get close to that limit!