In an attempt to avoid having to do laundry often, Marcus comes up with a system. He owns 10 shirts that are stored in his dresser. Every night, he designates his dirtiest shirt and throws it on his chair. In the morning, he randomly picks one of his shirts to wear (the ones in the dresser and the one on the chair). If he picked up the dirtiest one, he puts it in the dirty pile at the end of the day, and designates one of his remaining shirts as the current dirtiest one. He does his laundry once he’s out of shirts.
(a) What is the expected number of days it takes for him to do his laundry?
(b) Say he gets even lazier, and instead of organizing his shirts as he had planned, on each day he throws his shirts randomly into one of 10 different locations in his room (one shirt per location). Each day, he designates one of his shirts as his dirtiest shirt and one location as the dirtiest location. In the morning, if he happens to pick the dirtiest shirt, and the dirtiest shirt was in the dirtiest location, then he puts the shirt into the dirty pile and does not use that location anymore (it is too dirty now). What is the expected number of days for him to do his laundry now?