Page 417 - Discrete Mathematics and Its Applications

396 6 / Counting

Exercises

1. There are 18 mathematics majors and 325 computer sci- 17. How many strings of five ASCII characters contain the
ence majors at a college. character @ (“at” sign) at least once? [Note: There are
a) In how many ways can two representatives be picked 128 different ASCII characters.
so that one is a mathematics major and the other is a 18. How many 5-element DNA sequences
computer science major? a) end with A?
b) In how many ways can one representative be picked b) start with T and end with G?
who is either a mathematics major or a computer sci- c) contain only A and T?
ence major? d) do not contain C?
2. An office building contains 27 floors and has 37 offices 19. How many 6-element RNA sequences
on each floor. How many offices are in the building? a) do not contain U?
3. A multiple-choice test contains 10 questions. There are b) end with GU?
four possible answers for each question. c) start with C?
a) In how many ways can a student answer the questions d) contain only A or U?
on the test if the student answers every question? 20. How many positive integers between 5 and 31
b) In how many ways can a student answer the questions a) are divisible by 3? Which integers are these?
on the test if the student can leave answers blank? b) are divisible by 4? Which integers are these?
4. A particular brand of shirt comes in 12 colors, has a male c) are divisible by 3 and by 4? Which integers are these?
version and a female version, and comes in three sizes 21. How many positive integers between 50 and 100
for each sex. How many different types of this shirt are a) are divisible by 7? Which integers are these?
made?
b) are divisible by 11? Which integers are these?
5. Six different airlines fly from New York to Denver and c) are divisible by both 7 and 11? Which integers are
seven fly from Denver to San Francisco. How many dif- these?
ferent pairs of airlines can you choose on which to book 22. How many positive integers less than 1000
a trip from NewYork to San Francisco via Denver, when
a) are divisible by 7?
you pick an airline for the flight to Denver and an airline
b) are divisible by 7 but not by 11?
for the continuation flight to San Francisco?
c) are divisible by both 7 and 11?
6. There are four major auto routes from Boston to Detroit d) are divisible by either 7 or 11?
and six from Detroit to Los Angeles. How many major e) are divisible by exactly one of 7 and 11?
auto routes are there from Boston to Los Angeles via De- f) are divisible by neither 7 nor 11?
troit? g) have distinct digits?
7. How many different three-letter initials can people have? h) have distinct digits and are even?
23. How many positive integers between 100 and 999 inclu-
8. How many different three-letter initials with none of the
sive
letters repeated can people have?
a) are divisible by 7?
9. How many different three-letter initials are there that be- b) are odd?
gin with an A?
c) have the same three decimal digits?
10. How many bit strings are there of length eight? d) are not divisible by 4?
11. How many bit strings of length ten both begin and end e) are divisible by 3 or 4?
with a 1? f) are not divisible by either 3 or 4?
g) are divisible by 3 but not by 4?
12. How many bit strings are there of length six or less, not
counting the empty string? h) are divisible by 3 and 4?
24. How many positive integers between 1000 and 9999 in-
13. How many bit strings with length not exceeding n, where clusive
n is a positive integer, consist entirely of 1s, not counting a) are divisible by 9?
the empty string?
b) are even?
14. How many bit strings of length n, where n is a positive c) have distinct digits?
integer, start and end with 1s? d) are not divisible by 3?
15. How many strings are there of lowercase letters of length e) are divisible by 5 or 7?
four or less, not counting the empty string? f) are not divisible by either 5 or 7?
g) are divisible by 5 but not by 7?
16. How many strings are there of four lowercase letters that
have the letter x in them? h) are divisible by 5 and 7?