Xuan Guo xguo9@student.gsu.edu Lab 9 Xuan Guo xguo9@student.gsu.edu

Question 5 Section 6.1 Six different airlines fly from New York to Denver and seven fly from Denver to San Francisco. How many different pairs of airlines can you choose on which to book a trip from New York to San Francisco via Denver, when you pick an airline for the flight to Denver and an airline for the continuation flight to San Francisco?

Question 5 Section 6.1 Six different airlines fly from New York to Denver and seven fly from Denver to San Francisco. How many different pairs of airlines can you choose on which to book a trip from New York to San Francisco via Denver Solution: 42 The Product Rule: A procedure can be broken down into a sequence of two tasks. There are n1 ways to do the first task and n2 ways to do the second task. Then there are n1∙n2 ways to do the procedure. 6 x 7

Question 29 Section 6.1 How many license plates can be made using either two uppercase English letters followed by four digits or two digits followed by four uppercase English letters?

Question 29 Section 6.1 How many license plates can be made using either two uppercase English letters followed by four digits or two digits followed by four uppercase English letters? Solution: 52,457,600 Product Rule 26 x 26 x 10 x 10 x 10 x 10 10 x 10 x 26 x 26 x 26 x 26

Question 33 Section 6.1 How many strings of eight English letters are there a) that contain no vowels, if letters can be repeated? b)that contain no vowels, if letters cannot be repeated? c) that start with a vowel, if letters can be repeated? d)that start with a vowel, if letters cannot be repeated? e) that contain at least one vowel, if letters can be repeated? f) that contain exactly one vowel, if letters can be repeated? g) that start with X and contain at least one vowel, if letters can be repeated? h)that start and end with X and contain at least one vowel, if letters can be repeated?

Question 33 Section 6.1 How many strings of eight English letters are there a) that contain no vowels, if letters can be repeated? 37,822,859,361 = (26 – 5) ^ 8 b)that contain no vowels, if letters cannot be repeated? 8,204,716,800 = 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 c) that start with a vowel, if letters can be repeated? 40,159,050,880 = 5 x (26^7) d)that start with a vowel, if letters cannot be repeated? 12,113,640,000 = 5 x 26 x 25 x 24 x 23 x 22 x 21 x 20 e) that contain at least one vowel, if letters can be repeated? 171,004,205,215 = (26^8) – (21^8) f) that contain exactly one vowel, if letters can be repeated? 72,043,541,640 = 5 x (21^7) x 8 g) that start with X and contain at least one vowel, if letters can be repeated? 6,230,721,635 = (26^7) – (21^7) h)that start and end with X and contain at least one vowel, if letters can be repeated? 223,149,655 = (26^6) – (21^6)

Question 1 Section 6.2 Show that in any set of six classes, each regularly once a week on a particular day of the week, there must be two that meet on the same day, assuming that no classes are held on weekends.

Pigeonhole Principle: If k is a positive integer and k + 1 objects are placed into k boxes, then at least one box contains two or more objects.

Question 1 Section 6.2 Show that in any set of six classes, each regularly once a week on a particular day of the week, there must be two that meet on the same day, assuming that no classes are held on weekends. Solution: Because there are six classes, but only five weekdays, the pigeonhole principle shows that at least two classes must be held on the same day.

Question 3 Section 6.1 A drawer contains a dozen brown socks and a dozen black socks, all unmatched. A man takes socks out at random in the dark. a) How many socks must he take out to be sure that he has at least two socks of the same color? b)How many socks must he take out to be sure that he has at least two black socks?

Question 3 Section 6.1 A drawer contains a dozen brown socks and a dozen black socks, all unmatched. A man takes socks out at random in the dark. a) How many socks must he take out to be sure that he has at least two socks of the same color? Solution: 3 Pigeonhole Principle: If k is a positive integer and k + 1 objects are placed into k boxes, then at least one box contains two or more objects.

Question 3 Section 6.1 A drawer contains a dozen brown socks and a dozen black socks, all unmatched. A man takes socks out at random in the dark. b)How many socks must he take out to be sure that he has at least two black socks? Solution: 14