#1

   Consider the word  expenseless .

 (1)  Assuming that identical letters are indistinguishable, how many unique rearrangements are there of the letters in the word  ?

  (2)  How many ways can the letters in the word be rearranged that result in no detectable difference in the resulting arrangement, assuming that identical letters are indistinguishable?

  (3)  How many ways are there to arrange these letters so that no vowels are adjacent?

#2

John manages an RS6000 computer with a tape drive. Thursday he was given a back-up tape that contained 500,000 "words," each word created from four or fewer lowercase letters and each word separated by a single blank character.  

 Argue that at least one of the "words" must be repeated among the 500,000 entries on the tape.

#3

 Focus on the expansion of

 (a + b + c + d + e + f + g)^26.

(1)   Determine the number of uncollected terms in the expansion.  

(2)  Determine the number of collected terms in the expansion.   

(3)  Determine the coefficient K for the collected term

        K a^2b^4c^6d^10eg^3.

 (4)  How many collected terms are of the form                      C  m^k n^(26-k),    m,n  from among       {a,b,c,d,e,f,g} ?

#4

 Pat Punchlormer is a friend of Sandy Softknuckle's. Pat obsesses about the myriad ways to create work outfits comprised of a shirt, a pair of slacks, and a pair of shoes.

  (1)  From a closet with 20 different shirts, 15 different pairs of slacks, and 10 different pairs of shoes, how many different outfits can be created?  

   (2)  If 6 different shirts and 8 different pairs of slacks are available, how many different pairs of shoes are required so that a different outfit can be worn each day during a typical working year?

#5

A survey was taken of Chicago-area businesses to determine the number of fulltime permanent employees maintained in each business. Among other requests, respondents were asked to respond to the item below. Exactly 1240 businesses responded to the survey.

Our business maintains (check one):
+ less than 26 employees
+ 26-49 employees
+ 50-99 employees
+ 100-999 employees
+ 1000 or more employees

 (1)  How many different ways could the group of 1240 businesses respond to the item shown above?

 (2)  If no less than 10% of the businesses are in each employee categorization shown in the box, how many different ways could the group of businesses respond to the item?

  (3) If the response to this item is recorded and identified for each of the 1240 businesses, how many different sets of responses are possible?

#6

 

Gauss's Gentle Grove is a romantic restauranton the outskirts of the Black Forest. Gauss has the same 32 varieties of wine to choose from each night, and each night's wine list has exactly 8 entries.  

(1) How many different wine lists can be created? Solve the problem assuming the order of appearance on the list does not matter.

 (2) If 20 of the wines are red and 12 are white, how many ways can a list be created to include 5 reds and 3 whites? Use the same order assumption as for the previous problem.

(3) At the Grove, as the locals call the restaurant, Gauss has 28 German wines and 4 French wines. If an evening's wine list includes no more than one French wine, how many different wine lists can be written? Use the same order assumption as for the previous problem.

(4) Fourteen of Gauss's wines are dry and 18 are semi-dry. If no dry wine is listed as the first or last wine on an evening's list, how many different wine lists can be written?

 (5) If each of the 32 wines can be classified as red or white, German or French, and dry or semi-dry, can an evening's list contain wines each of a different classification?

 (6) Suppose we know that 12 of Gauss's 32 wines are red German semi-dry wines. Determine how many varieties there are of each of the other wine types, or alternatively, argue that more information is needed.

#7

  

My cousin Seth has a modest collection of football cards from the 28 professional NFL teams. He selects one card from each team. No two players are the same.

(1)  After Seth has selected one card from each team, how many ways can he arrange them in a single line?

(2)   How many ways can Seth arrange the 28 cards in a circle?

(3)   If 16 of the cards represent defensive players, 10 represent offensive players, and 2 represent place kickers, how many ways are there to arrange the 28 cards in a single line so that all players of the same group (defense, offence, kicker) are together?

 (4)  Seth decided to categorize the 28 players according to the home state of residence listed on the back of the card. Assuming all players are from the United States, how many different categorizations are there?

#8

Write an inductive proof to justify the following conjecture :

 The sum of the first n terms of the form   (2)(3^(j-1)),

j ranging from 1 to n,  is  3^n - 1.

#9

 Write the next 6 terms in the following recursion relation.

a(n) = 2a(n-1) - 3a(n-2),

a(0 )= 2, a(1) = 5

#10

 Write both a recursive and an explicit representation for the following sequence of values.

                              5, 17,  41,  89,  185