Why Read This Book? - Index of

4.8 Cartesian Products and Cardinality 149
EXERCISE 4.8.13 How many meals can be put together from the menu in
Example 4.8.12 if dessert is optional?
Here is an illustration of the practical use of Theorem 4.8.8.
Example 4.8.14 Suppose you toss a coin 10 times and observe the sequence of
outcomes of heads (H) or tails (T). Each possible outcome of the 10 tossings can
be written as a 10-tuple of the form (H,H,H,T,H,H,T,T,H,T), each of which we can
visualize as a function from N10 to {H,T}. By Theorem 4.8.8, there are 2 10 = 1024
such functions. �
EXERCISE 4.8.15 A password to your computer account must be precisely
seven alphanumeric characters, that is, a sequence of seven characters taken from
the 26 letters of the alphabet and the digits 0–9. How many distinct passwords
may be formed?
Exercise 4.8.9 will help you answer the next question.
EXERCISE 4.8.16 The Fitzpatricks are expecting a baby boy any day now.
Family members have strong opinions about what the child will be named. Suggested
names are William, Warren, Benjamin, Fitzhugh, Chancellor, Millhouse,
and Nebuchadnezzar. The parents want their son to have three distinct given
names, such as William Fitzhugh Millhouse Fitzpatrick. How many potential
names are there for the Fitzpatrick son?
EXERCISE 4.8.17 How many ways are there to arrange your collection of eight
CDs in a row on top of your dresser?
EXERCISE 4.8.18 In California, a standard license plate consists of a digit 1–9,
followed by three letters, followed by three more digits 0–9, such as 3AAG045. If
all constructions of this form are considered usable, how many standard license
plates can California issue?
EXERCISE 4.8.19 Twenty horses run a race in which prizes are given for win,
place, and show (first, second, and third places). How many outcomes are there
for the race?
Example 4.8.20 Let a word be defined as any arrangement of letters from the
alphabet, so that, for example EIEIO is a word. If the vowels are {A, E, I, O, U},
how many four-letter words contain at least one vowel?
Solution Let U be the set of all four-letter words, and let A be the set of fourletter
words that contain at least one vowel. Then |U| = 26 4 and � � A C � � = 21 4 .By
the complement rule, |A| = 26 4 − 21 4 . �