Generally, stocking a bar is a pretty simple affair. You get a big bottle of your basic alcohols (rum, gin, vodka, whiskey, tequila), smaller bottles of some other stuff (vermouth, bitters, etc), and a bunch of mixers.
Given a well-stocked bar, you can make a lot of drinks. However, you can’t make all the drinks, or rather, you shouldn’t. Not all combinations of ingredients will work well together. A rum and Coke is fine. A Coke and Coke won’t impress anyone, and a Midori and Coke would be awful, so the upper bound on the number of drinks you can make with a set of N alcohols is much lower than pure math would suggest.
I’m very gradually building a barbot, which adds a further complication: the robot only has five pumps. So from all the available alcohols, mixers, and so on, I have to choose five (there’s also a practical concern, which is that pumping soda shakes it up badly, but I’m choosing to ignore that for now).
For example, gin, tonic, vodka, orange juice, and cranberry juice would let me make a Vodka and Tonic, G&T, Screwdriver, Gin & Juice, and a Cape Cod. That’s pretty good, with 5 different drinks available from 5 different ingredients (10 if you count “shot of gin”, “glass of orange juice”, and so on as drinks).
But I want to know what the set of five liquors with the most possible mixed drinks is. To that end, I’ve downloaded the complete set of mixed drinks from the webtender, which I plan to use as the data for making my drink set.
The algorithm is another matter.