Section 4.2 Counting
261
Exercises 33–42 concern the set of binary strings of length 8 (each character is either the digit 0 or the digit 1).
33. How many such strings are there?
34. How many begin and end with 0?
35. How many begin or end with 0?
36. How many have 1 as the second digit?
37. How many begin with 111?
38. How many contain exactly one 0?
39. How many begin with 10 or have a 0 as the third digit?
40. How many are palindromes? (See Exercise 11.)
41. How many contain exactly seven 1s?
42. How many contain two or more 0s?
In Exercises 43–48, 2 dice are rolled, 1 black and 1 white.
43. How many different rolls are possible? (Note that a 4-black, 1-white result and a 1-black, 4-white result
are two different outcomes.)
44. How many rolls result in “snake eyes” (both dice showing 1)?
45. How many rolls result in doubles (both dice showing the same value)?
46. How many rolls result in a total of 7 or 11?
47. How many rolls occur in which neither die shows the value 4?
48. How many rolls occur in which the value on the white die is greater than the value on the black die?
In Exercises 49–54, a customer is ordering a new desktop computer system. The choices are 21-inch, 23-inch,
or 24-inch monitor (optional); 1 TB or 2 TB hard drive; 6 GB or 8 GB of RAM; 16X DVD or Blu-ray disk
optical drive; Intel, AMD, or NVIDIA video card; inkjet, laser, or laser color printer; 1-, 2-, or 3-year warranty.
49 How many different machine configurations are possible?
50. How many different machines can be ordered with a 2 TB hard drive?
51. How many different machines can be ordered with a 21-inch monitor and an inkjet printer?
52. How many different machines can be ordered if the customer decides not to get a new monitor?
53. How many different machines can be ordered with a 21-inch monitor, a 1 TB hard drive, and an inkjet
printer?
54. How many different machines can be ordered if the customer does not want a 3-year warranty?
Exercises 55–58 refer to the Konami code, a sequence of 10 keystrokes using 6 different characters:
c c T T d S d S B A. This code was inadvertently left by the developer in the first release of a video game;
when entered on the video game console as the title screen is open, it surreptitiously adds assets to the player’s
on-screen avatar. This code or versions of it have been retained in many video games.
55. How many 10-character codes can be created using these 6 characters?
56. How many 10-character codes can be created if the BA sequence must be the last 2 characters?
57. How many 10-character codes can be created if only 2 down-arrow characters can be used and they must
be paired together wherever they appear?
58. How many 10-character codes can be created if only the arrow characters are used?
261
Exercises 33–42 concern the set of binary strings of length 8 (each character is either the digit 0 or the digit 1).
33. How many such strings are there?
34. How many begin and end with 0?
35. How many begin or end with 0?
36. How many have 1 as the second digit?
37. How many begin with 111?
38. How many contain exactly one 0?
39. How many begin with 10 or have a 0 as the third digit?
40. How many are palindromes? (See Exercise 11.)
41. How many contain exactly seven 1s?
42. How many contain two or more 0s?
In Exercises 43–48, 2 dice are rolled, 1 black and 1 white.
43. How many different rolls are possible? (Note that a 4-black, 1-white result and a 1-black, 4-white result
are two different outcomes.)
44. How many rolls result in “snake eyes” (both dice showing 1)?
45. How many rolls result in doubles (both dice showing the same value)?
46. How many rolls result in a total of 7 or 11?
47. How many rolls occur in which neither die shows the value 4?
48. How many rolls occur in which the value on the white die is greater than the value on the black die?
In Exercises 49–54, a customer is ordering a new desktop computer system. The choices are 21-inch, 23-inch,
or 24-inch monitor (optional); 1 TB or 2 TB hard drive; 6 GB or 8 GB of RAM; 16X DVD or Blu-ray disk
optical drive; Intel, AMD, or NVIDIA video card; inkjet, laser, or laser color printer; 1-, 2-, or 3-year warranty.
49 How many different machine configurations are possible?
50. How many different machines can be ordered with a 2 TB hard drive?
51. How many different machines can be ordered with a 21-inch monitor and an inkjet printer?
52. How many different machines can be ordered if the customer decides not to get a new monitor?
53. How many different machines can be ordered with a 21-inch monitor, a 1 TB hard drive, and an inkjet
printer?
54. How many different machines can be ordered if the customer does not want a 3-year warranty?
Exercises 55–58 refer to the Konami code, a sequence of 10 keystrokes using 6 different characters:
c c T T d S d S B A. This code was inadvertently left by the developer in the first release of a video game;
when entered on the video game console as the title screen is open, it surreptitiously adds assets to the player’s
on-screen avatar. This code or versions of it have been retained in many video games.
55. How many 10-character codes can be created using these 6 characters?
56. How many 10-character codes can be created if the BA sequence must be the last 2 characters?
57. How many 10-character codes can be created if only 2 down-arrow characters can be used and they must
be paired together wherever they appear?
58. How many 10-character codes can be created if only the arrow characters are used?