motel rooms

I5 Motel Rooms

A motel has ten rooms, all located on the same side of a single corridor and numbered 1 to 10 in numerical order. The motel always randomly allocates rooms to its guests. There are no other guests besides those mentioned. a Friends Molly and Polly have been allocated two separate rooms at the motel. What is the most likely number of rooms between their rooms? b Molly believes there is a greater than one-third chance that at most

one room will separate them, but Polly disagrees. Who is right? Ex- plain why.

c On another occasion, Molly, Polly, and a third friend, Ollie, were allocated three separate rooms. Molly believes there is a better than one-third chance they are all within a block of five consecutive rooms. Ollie believes there is exactly one-third chance and Polly believes there is less than one-third chance. Who is right? Explain why. d Ollie arrived after rooms were allocated to Molly and Polly. There was then a 50% chance he would be in a room adjacent to Molly or Polly or both. In how many ways could a pair of rooms have been allocated to Molly and Polly?