448 CHAPTER 7 Counting and Combinatorics
U Exercises
- A "word" is a string of one or more lowercase letters. How many words can be formed
using all the letters of the word hyperbola? In how many words will h and y occur
together? In how many will h and y not occur together? - A bookshelf contains three novels, six books of poetry, and four reference books.
In how many ways can these books be arranged so that the books of each type are
together? - A shelf has room for 10 books.
(a) Given an inventory of 25 books, how many years will it take to display all combi-
nations of 10 books if the display is changed once a week?
(b) How many years will it take if the display is changed five times a week? - A passenger train consists of two baggage cars, four day coaches, and three parlor
cars. In how many ways can the train be made up if the two baggage cars must be in
the front and the three parlor cars must be in the rear? Assume that the baggage cars
can be told apart, that the day coaches can be told apart, and that the parlor cars can be
told apart. - The English alphabet contains 26 letters, including five vowels. In each case determine
how many words of length five are possible provided that:
(a) Words contain at most two distinct vowels
(b) Words contain at most one letter that is a vowel
(c) Words contain at least four distinct vowels - How many five-letter words formed with a, b, and c have at least one letter missing?
- A student has four examinations to write, and there are 10 examinations periods avail-
able. How many ways are there to schedule the examinations? - A student must complete the following sequence of courses: Two of four lab science
courses, one of two literature courses, two of three mathematics courses, and one of
seven physical education courses. Assume that none of these courses is a prerequisite
for any other.
(a) How many ways can courses be chosen if the possibility of time conflicts is
disregarded?
(b) How many ways can courses be chosen if two different lab courses are scheduled
at the same time as one of the literature courses?
(c) How many ways can courses be chosen if all the physical education courses are
offered at the same time as one of the literature courses? - A student must answer 8 out of 10 questions on an exam.
(a) How many choices does the student have?
(b) How many choices does a student have if the first three questions must be
answered?
(c) How many choices does a student have if exactly four out of the first five questions
must be answered? - How many ways can you list the 12 months of the year so that May and June are not
adjacent?