Note.Whenever you use the Multiplication Rule (MR) or the Addition Rule in your solutions you have to state this explicitly: e.g., "by the multi- plication rule or by MR". You must also persuade yourselves and explicitly state that the number of ways in which each step in the MR is performed does not depend on how the previous steps were performed.

1. In a country called Oddia the phone numbers consist of a 2-digit area code followed by a 4-digit personal number: (xx)yyyy. The phone numbers are subject to the following rules: (1) the area code cannot begin with 0,8 or 9, (2) the personal numbers cannot begin with 5 and must be odd, (3) the phone numbers made with the same digit in all 6 positions are reserved for Oddia's Presedinte. How many possible phone numbers are available for Oddia's ordinary residents (who are not the Presedinte)? (You can assume, and do not need to prove, that a multi-digit number is odd exactly when its last digit is odd.)

2. LetA,Bbe two finite sets and denotem=|A|,n=|B|. Compare|2A2B|and|2AB|(state which is strictly biger or maybe they are equal) for various natural number values ofmandn. There are only a few cases, which you must, however, identify. You do not need to prove anything, just state the answer for each case.

3.n2 distinguishable Hogwarts students participate in Profes- sor Snape's experiment. Each student is given one of three concoctions: potion A, or potion B, or a mixture of the two. Snape makes sure to give a different concoction to each of Harry and Hermione. In how many distinct ways could Snape have distributed his concoctions?

4. Prove that each of the following integers is composite (not prime).

(a) 16991 (b) 366+ 4(1333)

5. We stipulate that pair of integers (x, y) is a pair ofBroadway integersif the difference between the square ofxandy2is divisible by 23. Prove that every pair of odd integers is a pair of Broadway integers.