Count Items in an Array of Arrays?

You can add the nested array counts with

let count = compoundArray.reduce(0) { $0 + $1.count }

Performance comparison for large arrays (compiled and run on a MacBook Pro in Release configuration):

let N = 20_000
let compoundArray = Array(repeating: Array(repeating: "String", count: N), count: N)

do {
    let start = Date()
    let count = compoundArray.joined().count
    let end = Date()
    print(end.timeIntervalSince(start))
    // 0.729196012020111
}

do {
    let start = Date()
    let count = compoundArray.flatMap({$0}).count
    let end = Date()
    print(end.timeIntervalSince(start))
    // 29.841913998127
}

do {
    let start = Date()
    let count = compoundArray.reduce(0) { $0 + $1.count }
    let end = Date()
    print(end.timeIntervalSince(start))
    // 0.000432014465332031
}

You can use joined or flatMap for that.

Using joined

let count = compoundArray.joined().count

Using flatMap

let count = compoundArray.flatMap({$0}).count

Since you are asking for a property, I thought I'd point out how to create one (for all collections, while we're at it):

extension Collection where Iterator.Element: Collection {
    var flatCount: Int {
        return self.reduce(0) { $0 + $1.count } // as per Martin R
    }
}

Making this recursive seems to be an interesting exercise.