You just rented a large house and the realtor gave you 5 keys, one for each of the 5 doors of the house. Unfortunately, all keys look identical, so to open the front door, you try them at random. (a) Define the random variable X equal to the number of keys you will need to try to open the door. Find the probability mass function of X under the following assumption: after an unsuccessful trial you mark the corresponding key so that you never try it again. (b) Repeat part (a) for the case where the realtor gave you an extra duplicate key for each of the 5 doors. Hint: define Ki as the event that the i-th key selected opens the door.