Module 08 Written Assignment - 1. Prove that the following set is countably infinite: a {q Q: q= where a is evenbis odd } b 2. In a class of 50 students, how many students are guaranteed to get the same score on an equally-weighted 20 question quiz? (Assume each question is either completely right or completely wrong - no partial credit.) 3. A man has 10 black socks, 11 brown socks and 12 blue socks in a drawer. He isn't a morning person, so every morning he just reaches in and pulls out socks until he gets two that match. a. Use the generalized pigeonhole principle to determine how many socks he has to pull out before he gets two of the same color. b. How many must he pull out to be certain of getting a blue pair? a. b. Module 09 Written Assignment - 1. IP Addresses are stored as binary numbers in four 8-bit bytes (strings of 0's and 1's). For ease of reading, these are translated into their four equivalent decimal numbers, separated by dots, to form \"dot separated decimals.\" Use the facts given about binary numbers, along with the multiplication principle, to answer the following questions about IP addresses: a. What is the decimal value of the binary number 10101011? (Note that this is the same number shown in the place-value table below.) b. How many different decimal values are possible for an 8-bit binary number? c. How many different IP addresses can be stored as described above? Hints: Part b is really asking in how many ways a string of eight 0's and 1's can be arranged. Also, remember that the place value of binary numbers uses powers of 2, as shown below: Place Value of Binary Digits Place value 128's 64's 32's 16's 8's 4's 2's 1's 7 6 5 4 3 2 1 0 Associated power of 2 2 2 2 2 2 2 2 2 Binary digit 1 0 1 0 1 0 1 1 a. b. c. 2. A particular online banking system uses the following rules for its passwords: a. Passwords must be 6-8 characters in length b. Passwords must use only alphabetical and numeric characters, and must have at least one alpha and one numeric character. c. Letters are case sensitive. Under these rules, how many different passwords are possible? Give your answer to as many digits of accuracy as you can achieve on your calculator. a. b. c. 3. A club has 24 members. a. In how many ways can 4 officers, a president, vice-president, secretary and treasurer be chosen from the members of the club? b. In how many ways can a 4-person committee be chosen from the members of the club? a. b. 4. A group has 7 women and 8 men. They need to choose a delegation that includes 2 men and 2 women. a. In how many ways can this be done? b. In how many ways can it be done if ex-spouses Jennifer and Brad refuse to serve on the delegation together? a. b. 5. An ice cream shop has 31 different flavors of ice cream and two kinds of cones. The rules are, cones can be one, two or three-scoops (but no larger), and if two scoops are the same kind of ice cream they are indistinguishable. Otherwise the order of the scoops matters. How many different ice cream cones can a customer order at this shop