Qualifer Practice Exam: Foundations

Question 1 (3 Points): Conte di Savoia on Taylor Street (between Racine and Ashland) sells 6 different kinds of pastries. How many different pastry trays can they make that have 20 pastries, with at least 3 of those 20 being cannoli? Note that the order of the pastries on the tray does not make a tray different, but having different numbers of pastries does. (I.e., “10 apple strudels and 10 cannoli” is the same tray as “10 cannoli and 10 apple strudels”, but “11 apple strudels and 9 cannoli” is different. You may use exponents and factorial symbols in your answer; you do not need to simplify to a single integer.

Order doesn't matter, so this is a combinations problem. Now 3 of the twenty positions are already handled since 3 have to be cannoli. The formula for combinations with repetition is (n+k-1)!/k!(n-1)! Since ALL of the combinations will have 3 cannoli, we are only choosing 17 items of the 6 types. n=17, k=6. (22)!/6!16! (The problem says simplifying is unnecessary, but let's finish for the sake of cleanliness.

(22)!/6!16! = (17*18*19*20*21*22)/2*3*4*5*6 = 17*3*19*7*11 = 74613 different plates.

Question 2 (5 Points): Suppose there are 10 students in a class and that each will receive one of the following grades: A, B, C, D, F. For each of the following you need not give a final numerical answer; an equivalent expression is sufficient.

Part I: How many distinct ways are there for the instructor to assign the grades?

Now order matters, because student 1 is not student 2. So there are 5 grades being chosen (n), ten times (r). Therefore, there are 5^10 distinct ways to assign the grades.

Part II: Now suppose the Dean has stated that at least two of the students must receive F’s. With this added constraint on the instructor, how many distinct grade assignments are there?

So the tricky thing I'm not sure about here is that the core part is definitely 5^8, in that the eight students without F's can have any grade. Come back.