1) A certain western state produces license plates for all the registered vehicles in the state. Each license plate contains a field with four numbers, followed by a field with 3 letters of the alphabet. Numbers (1 to 9) are allowed to repeat; zero is allowed only in last cell of numeric field. Letters of the alphabet (A to Z) in alpha field must be distinct, that is, letters on license plate are not allowed to repeat. Letter \"O\" is not used by the state on license plates due to possible confusion with 'zero'. e How many different license plates can be produced by this western state? Show work! Evaluate your solution. 2) The owner of a dealership that sells unique vintage automobiles is planning to make a television commercial to feature his luxury vintage vehicles. Assume that the dealership has a showroom with 8 rotating platforms of different heights on which the vehicles can be prominently displayed. The dealership has a current inventory of nineteen unique (one-of-a-kind) luxury vintage automaobiles. The owner must select from his inventory the eight vintage vehicles that will be featured in the TV commercial, displayed on the rotating platforms in his showroom. In how many different ways may the owner of the dealership select the vintage vehicles to feature in the TV commercial on the rotating platforms? Show work! Evaluate your solution. 3) Assume that a history Quiz consists of nine multiple choice questions. Each question on the Quiz has six choices for the answer: A, B, C, D, E or \"none of the above\". Only one possible choice per answer is correct. A student taking the Quiz must select only one answer per question, and no answer may be left blank. Assume that student taking the history Quiz did not study at all, but is simply guessing at all the answers. a) In how many ways can the student randomly select all their answers on history Quiz? Show work. Evaluate your solution. b) In how many ways can the student randomly select all their answers on history Quiz, such that all of their answers are correct? Show work. Evaluate your solution. b) In how many ways can the student randomly select all their answers on history quiz, such that all of their answers are incorrect (wrong)? Show work. Evaluate your solution. 4) A local fire department employs a business manager who receives an annual salary. The local fire department has a total of 30 volunteer firefighters; they are not paid firefighters. Nineteen of the local volunteer firefighters are employed in local businesses. Eleven of the local volunteer firefighters are retired from FDNY as career firefighters; they are not employed locally. Nassau County has asked the business manager of the local fire department to select 8 local volunteers firefighters to attend an annual Firefighters Convention in upstate New York, at Lake Placid. All travel expenses will be paid by the County. a) Assume that all of the local volunteer firefighters are eligible to attend the Convention at Lake Placid. The selection will be made at random by the business manager in the fire department. In how many possible ways can any eight of the local volunteer firefighters be randomly selected to attend the Convention at Lake Placid? Express your solution symbolically and then evaluate. b) Now consider: In how many possible ways can 8 of the local volunteer firefighters be randomly selected to attend the Convention, such that one-half of those selected are retired from FDNY and the other half of those selected are employed in local businesses? Express your solution symbolically and then evaluate