Question. Solve each question CORRECTLY

I: I You come highly recommended to us from Hotel Gemini, and now your application for employment at Hotel Alofoyo is nearly complete. You know us as The Milky Way Galaxy's most popular hotel and resort accommodation. No doubt you are aware of our hotel's numerous tourist attractions, as we have been advertising in print and news mEdia outlets for many years now. In fact, we have lost count of both the number of attractions and the number of years we have been advertising. Nevertheless, you will recall that we have a countably innite number of rooms, and they are numbered 1, 2, 3, and so on. Also, we will remind you of our hotel's strict room policies: A. No arriving guest or current guest is ever turned away (it's the law). Every guest must be booked into a room every night (no sleeping in the hallways). . No guest may stay in more than one room on the same night [one room per guest). . No room may contain more than one guest (one guest per room). Every room must be lled every night (we need to make money). We always refer to guests by their current room number (guest X stays in room X). . Guests may be asked to change rooms at most once per day {happy guests). EDITJEUOPJ . All room changes must occur before anyone can check-in. I. A room registry must be completed every night [see below). For example, suppose that in one day (i) guests 2? and 3? check-out, (ii) guests 20 and 30 are asked to move into rooms 2? and 3? (in that order), and (iii) the rst and second arriving guests are assigned to rooms 20 and 30 (in that order). Then guest 20 becomes guest 2?, guest 30 becomes guest 3?, the rst arriving guest becomes guest 20, and the second arriving guest becomes guest 30. We always use the most compact notation possible to indicate these events on the room registry. In this case, we can represent the room assignments for returning guests as t : H - {27,37} -r N with 27, m = 20 rst) = 37, m. = 30 m, m. s 20,30 and we can represent the incoming guest room assignments as a : {1, 2} -+ N with 20, at=1 MI): {30 :r= 2 Note that the domain of d: always includm all returning guests [whether they change rooms or not} and that the domain of it? includes all newly-arriving guests numbered in the order they arrive. If multiple guests arrive at the same time, then you are free to choose which is the rst guest, which is the second guest, and so on. If you are unsure whether a room registry has been properly recorded, then nd a more experienced staff member to help you. (a) As you may have noticed by all the commotion around here, every room has again been booked for tonight. However, 100 dignitaries from Planet Injexon have suddenly arrived (unannounced, of course) and they are demanding to be given rooms for the night. Although I explained to them that we have none available, they rightly pointed out that we cannot legally refuse them rooms. My evening shift ends soon, but if you can help me gure out a way to assign rooms so that the hotel's policies are strictly adhered to, then I will give you the job tonight. (b) Welcome to the staff of Hotel Alefaya! On any given day, it is to be expected that is) some guests will complete their stay and check-out at noon. However, today we have some very bad luck because the only guests who are not leaving are those whose room numbers are given by factorial numbers. I am afraid that we will have so many empty rooms! For example, there will be a block of nearly 60!] empty rooms between rooms 5! 2 12d and ti! = 720 (and you can imagine even larger blocks of empty rooms elsewhere in the hotel). To make matters worse, no new guests will be arriving tonight due to construction on the Interplanetary Bijerron Highway. Please, nd a way to handle this situation so that our policies are still strictly adhered to (or this will be your last night!) Well done moving those guests around last night you have earned a promotion to assistant manager! As you know, we are again fully booked for this evening because no one is checking out today. However, due to a bookkeeping mistake on our part [and this is why the previous assistant manager was red) the nearby Denumeron Galaxy will be holding a convention here tomorrow and one repre- sentative from each planet in their Galaxy will be arriving tonight! I don't have to remind you, I'm sure, that Denumeron has a countably infinite number of planets. Even our bosses oonsider this to be a very odd situation. Still, when the guests arrive, and I tell them we are full and so there is no way we can accommodate an innite number of additional guests, then you and I are going to be red for certain. The Denumeron guests will arrive soon, and they will do so one at a time. You had better think of a solution fast. And don't forget to adhere to the hotel's strict policies! 5. (1 Credit) Suppose we have two vectors v and w, and we want to determine their cross product - however, we do not know what the vectors v and w are explicitly. We do know the following: . The length of v is 4, and length of w is 6. . The angle between the two vectors is */3. . The two vectors lie in the plane r + 2y + 3z = 0. Is this enough information to tell us what v x w is? If not, how many possible answers are there, based only on these three bullet points? Answer these questions in a few sentences. 6. (2 Credits) Find an equation for the plane that contains the points P = (1, 0,3), Q = (2, -2,2), and R = (6,0, -2). Give your answer in the form or + by + cz = d, where a, b, c and d are scalars