Rum and Coke was one of my favorite drinks in my 20s. Once I began eating more healthily, it was very simple to swap a diet Coke in for the regular sugar-filled Coke!
The ingredients here are rather simple -
start with a few ice cubes
add 1oz Rum
Fill glass with diet coke
garnish with a slice of lime
That's it! Remember to use REGULAR rum in this - i.e. Bacardi rum or something like that. As soon as you start going for the flavored rums, you're introducing carbs.
Rum has zero carbs, and diet Coke has zero carbs. So this is a carb free beverage!
That being said, I wouldn't drink a ton of these. Not only is overdrinking of alcohol bad for you, but overdrinking of SODA is bad for you. I have many articles on this site talking about how the caffeine and other chemicals in soda cause damage to your body, including leeching calcium out of your bones. It's fine to have one on a fun night out when you'll be dancing and enjoying yourself, but if you're thinking about drinking 3 a night because they are "without carbs", I would switch to a glass or two of wine instead.