Please write your solutions on one side only of each piece of paper you use, and please begin each solution on a new sheet of paper. You may use factorial notation as well as combination and permutation notation where appropriate (i.e., there is no need to expand 56!).

You are to work alone on this test. You may not use anyone else's work nor may you refer to any materials as you complete the test. You may ask me questions.

Evaluation Criteria

You may earn up to 10 points on each of questions 1 through 6. For each question:

€ 6 points count toward a correct solution to the problem.

     I will evaluate the mathematics you use:

+ Is it accurate and appropriate?
+ Have you provided adequate justification?

€ 4 points count toward how you express your solution.

     I will evaluate how you communicate your results:

+ Is your solution clear and complete?
+ Have you expressed logical connections among components of your solution?

#1

  Every weekday at lunch, Felix eats one apple and only one apple. Felix buys his apples from a grocer who always has the same 8 varieties of apples available.

(a) If Felix purchases apples only from this grocer and he wants to have a different variety of apple each weekday, Monday through Friday, how many different "apple menus" can Felix create?

(b) What if Felix decides it's okay to have the same variety of apple more than once during a week (M-F)? Now how many "apple menus" can he create?

#2

 

Behind the scenes at the upcoming 1996 Westminster Kennel Club Show in Madison Square Garden, there will be a room that contains 6 English Cocker Spaniels and 12 Fox Terriers. In how many ways can the 6 Cockers be paired with 6 of the Terriers?

 

#3

   A bag contains 26 red tokens labeled a, b, . . . , z and 42 green tokens labeled 2, 4, . . . , 84.

(a) In how many ways can 20 tokens be selected from the bag?

(b) In how many ways can 20 tokens of one color be selected from the bag?

#4

  Let us define a "word" as any arrangement of one or more letters of the 26-letter alphabet. How many two-letter "words" can be formed if each word must have distinct letters and no word can contain two consecutive letters of the alphabet? For example, bb is not a word, nor are jk and kj.

#5

  A group of 24 married couples has gathered at a northern Minnesota smelt fry. How many individuals must be chosen from the group to ensure that at least two of the persons chosen are a married couple?

#6

  Show that for any six positive integers, there must be at least one pair whose absolute difference is divisible by 5. Your justification should include more than simply an example that illustrates the result.