How to generate all possible combinations?

Try this.

The general algorithm is to have a fromList containing the letters you haven't used yet and a toList that is the string you've built up so far. This uses recursion to build up all possible strings and adds them to the set when the length is 2 or greater:

func permute(fromList: [String], toList: [String] = [String](), var set: Set<String> = Set<String>()) -> Set<String> {
    if toList.count >= 2 {
        set.insert(toList.joinWithSeparator(""))
    }
    if !fromList.isEmpty {
        for (index, item) in fromList.enumerate() {
            var newFrom = fromList
            newFrom.removeAtIndex(index)
            set = permute(newFrom, toList: toList + [item], set: set)
        }
    }
    return set
}

permute(["A", "B", "C"])
// {"BA", "AC", "ABC", "AB", "BCA", "CB", "BC", "CAB", "ACB", "CA", "CBA", "BAC"}

permute(["A", "A", "B"])
// {"BA", "BAA", "AAB", "AB", "ABA", "AA"}

Faster Answer:

As @MartinR pointed out in his post, the solution above is a little slow because of all of the creation and copying of sets. I had originally written this using an inout variable for set, but changed it to the more functional interface to make it nice to call.

Here is my original (faster) implementation, plus I embedded it in a permute that takes just an [String] and returns a Set<String>. It does the work of creating the set and the toList array and then calls the inner version of permute to do the real work:

func permute(list: [String], minStringLen: Int = 2) -> Set<String> {
    func permute(fromList: [String], toList: [String], minStringLen: Int, inout set: Set<String>) {
        if toList.count >= minStringLen {
            set.insert(toList.joinWithSeparator(""))
        }
        if !fromList.isEmpty {
            for (index, item) in fromList.enumerate() {
                var newFrom = fromList
                newFrom.removeAtIndex(index)
                permute(newFrom, toList: toList + [item], minStringLen: minStringLen, set: &set)
            }
        }
    }

    var set = Set<String>()
    permute(list, toList:[], minStringLen: minStringLen, set: &set)
    return set
}

permute(["A", "B", "C"])
// {"BA", "AC", "ABC", "AB", "BCA", "CB", "BC", "CAB", "ACB", "CA", "CBA", "BAC"}

permute(["A", "A", "B"])
// {"BA", "BAA", "AAB", "AB", "ABA", "AA"}

permute(["A", "A", "B"], minStringLen: 1)
// {"BA", "A", "BAA", "AB", "AA", "B", "AAB", "ABA"}

permute(["A", "A", "B"], minStringLen: 3)
// {"ABA", "BAA", "AAB"}

Edit: I added a minStringLen parameter (with default value of 2) instead of hard coding that value.

See @MartinR's answer for performance comparisons.

Swift 3 and Swift 4:

func permute(list: [String], minStringLen: Int = 2) -> Set<String> {
    func permute(fromList: [String], toList: [String], minStringLen: Int, set: inout Set<String>) {
        if toList.count >= minStringLen {
            set.insert(toList.joined(separator: ""))
        }
        if !fromList.isEmpty {
            for (index, item) in fromList.enumerated() {
                var newFrom = fromList
                newFrom.remove(at: index)
                permute(fromList: newFrom, toList: toList + [item], minStringLen: minStringLen, set: &set)
            }
        }
    }

    var set = Set<String>()
    permute(fromList: list, toList:[], minStringLen: minStringLen, set: &set)
    return set
}

print(permute(list: ["A", "B", "C"]))
// ["ABC", "CA", "BAC", "ACB", "BA", "CAB", "BC", "CB", "BCA", "CBA", "AB", "AC"]

print(permute(list: ["A", "A", "B"]))
// ["AA", "AAB", "ABA", "AB", "BA", "BAA"]

print(permute(list: ["A", "A", "B"], minStringLen: 1))
// ["AAB", "ABA", "B", "BA", "A", "BAA", "AA", "AB"]

print(permute(list: ["A", "A", "B"], minStringLen: 3))
// ["AAB", "ABA", "BAA"]