Advanced Algebra
Chapter 6 Answers
Counting and Arranging Discrete Objects
Mathematics can be used to help analyze language. Permutations and combinations are important tools for breaking codes, but they can also be helpful in solving simple word games.
Part A, Methods of Counting
1. The 8-letter words "equation," "dialogue," and "sequoia" are the three shortest common words containing the 5 common vowels. But the 8-letter scientific names of a certain beetle and fly contain 6 vowels (a, e, i, o, u, y), plus two different consonants.
a. The beetle was discovered in 1842, the fly in 1911. Here is the pattern of vowels (V) and consonants (C) in their names:
How many choices of vowels are there for V1? for V2? for each of the other vowels? [6, 5, 4, 3, 2, 1]b. How many choices of consonants are there for C1? for C2? [20, 19]
c. How many 8-letter words follow the vowel-consonant pattern of the two names? [273,600]
d. Find the names of the beetle and the fly. [Euryomia, Eumyobia]
Part B, Arrangements and Permutations
2. In a World Championship Scrabble match, your letter rack contains
these letters, all consonants. 
At the top of the board there's
a free
, to build a word around. You want to use all of your
letters on this turn. To do this you'll have to think of the word which,
in its plural form, is the longest word in the English language containing
only a single vowel.
a. How many permutations of the letters T G S N R H T E will you have to consider to find the only one you can use? [20,160]
b. What is the word? If you can't see it, consult an anagram engine. [STRENGTH]
Part C, Making Connections
3. Along with letter frequencies and word frequencies, double-letter patterns are extremely useful in cracking codes.
a. Find the most common double-letter patterns. [LL, EE, SS, OO, TT]
b. A biologist had to postpone an exhibit when her Gymnoti got sick. She hung a sign which, encoded in a substitution cipher, read MMPS BTT ZTT. How many 3-letter patterns containing a double letter are there? How many 4-letter words containing a double letter are there? [1300; 48,750]
c. What did the sign say? Find out what Gymnoti are. [EELS ALL ILL]
Each year, the National Institute of Standards and Technology, a U.S. government agency, presents the Malcolm Baldrige National Quality Award to one or more U.S. companies for excellence and quality achievement.
Part A, Combinations
1. Granite Rock Company won a Baldrige Award in 1992.
a. Write an expression nCr which could be evaluated to find the number of ways 5-member quality teams can be chosen from among all of Granite Rock's employees. [400C5]
b. The reliability of several Granite Rock processes has reached the six-sigma level. What is the probability that a given step in one such process will be defective? [0.0000034]
c. Solectron Corporation won a Baldrige Award in 1991. How many ways could the number of superior performance awards that the company won in 1990 be chosen from all of the awards it has won in the past 10 years? [37C10 = 3,628,800]
Part C, Making Connections
2 . Wainwright Industries won a Baldrige Award in 1994.
a. Find Wainwright's rate of overall customer satisfaction since 1992 and express it as a decimal. [0.95]
b. The probability that a number of customers in a group will be satisfied with a company's performance can be modeled by the binomial (dx + sy)n, where d is the rate of customer dissatisfaction, s is the rate of customer satisfaction, and n is the number of customers in the group. Write the binomial for a group of five Wainwright Industries customers. [(0.05x + 0.95y)5]
c. Expand the binomial. [0.0000003x5 + 0.0000297x4y + 0.0011281x3y2 +0.0214344x2y3 + 0.2036266xy4 + 0.7737809y5]
d. The probability that exactly a customers in a group of n customers are dissatisfied is given by the coefficient of xay n - a in the binomial expansion of (dx + sy) n. Find the probability that, in a randomly chosen group of five Wainwright customers, at least three are dissatisfied. [At least 3" means 3, 4, or 5 are dissatisfied; 0.0011281 + 0.0000297 + 0.0000003 = 0.0011581, or about one-tenth of 1 percent.]