3. Eight different people want to interview for the Defense Against the Dark Arts professorship at Hogwarts: Severus Snape, Tom Riddle, Remus Lupin, Dolores Umbridge, Mad-Eye Moody, Quirinus Quirrell, Barty Crouch, and Gilderoy Lockhart. Albus Dumbledore needs to interview the candidates. How many possible orders are there in which Dumbledore can interview the candidates if all of the following conditions must be simultaneously true? Snape must be interviewed before Riddle, Dolores Umbridge is interviewed last, • Crouch and Moody must be interviewed at the same time in a joint interview, AND • Lupin and Lockhart are interviewed consecutively (although we do not know which of the two is interviewed first)

Two new drugs are to be tested using a group of 60 laboratory mice, each tagged with a number for identification purposes. Drug A is to be given to 21 mice, drug B is to be given to another 21 mice, and the remaining 18 mice are to be used as controls. How many ways can the assignment of treatments to mice be made? (A single assignment involves specifying the treatment for each mouse—whether drug A, drug B, or no drug.) (Enter the exact number or an equivalent algebraic expression.)

A bagel shop has bagels in the following six varieties: plain, sesame, onion, blueberry, asiagocheese, and everything. Assuming there are at least 20 bagels of each variety, how many waysare there to choose 20 bagels...a. with no restrictions.b. with three or fewer blueberry bagels.c. with three or fewer blueberry bagels and three or fewer onion bagels.

The Tiny State College Student Advisory Board has rules to the effect that the officers of the board (i.e., Chairperson, Vice-Chairperson, and Treasurer) must always include at least one in-state student and at least one senior. The TSC Student Council consists of 6 members: 1 in-state senior, two in-state students who are not seniors, 1 out-of-state senior, and two out-of-state students who are not seniors. In how many ways can the officers of the Student Advisory Board be selected from these 6 Student Council members if they must obey the rules? Note that if individual members switch offices, this produces a different Board.