First create an NSDate for the given year and month:
let dateComponents = NSDateComponents()
dateComponents.year = 2015
dateComponents.month = 7
let calendar = NSCalendar.currentCalendar()
let date = calendar.dateFromComponents(dateComponents)!
Then use the rangeOfUnit() method, as described in
Number of days in the current month using iPhone SDK?:
// Swift 2:
let range = calendar.rangeOfUnit(.Day, inUnit: .Month, forDate: date)
// Swift 1.2:
let range = calendar.rangeOfUnit(.CalendarUnitDay, inUnit: .CalendarUnitMonth, forDate: date)
let numDays = range.length
print(numDays) // 31
Update for Swift 3 (Xcode 8):
let dateComponents = DateComponents(year: 2015, month: 7)
let calendar = Calendar.current
let date = calendar.date(from: dateComponents)!
let range = calendar.range(of: .day, in: .month, for: date)!
let numDays = range.count
print(numDays) // 31