4-4 Counting 187 12. Survey Cross Validation One way to identify survey subjects who don’t take the survey seriously is to repeat a question with similar wording. If a survey with 10 questions includes three questions that are the same except for minor differences in wording, how many different ways can the 10 questions be arranged? 13. Safety with Numbers The author owns a safe in which he stores all of his great ideas for the next edition of this book. The safe “combination” consists of four numbers, with each number from 0 to 99. The safe is designed so that numbers can be repeated. If another author breaks in and tries to steal these ideas, what is the probability that he or she will get the correct combination on the first attempt? Assume that the numbers are randomly selected. Given the number of possibilities, does it seem feasible to try opening the safe by making random guesses for the combination? 14. Electricity When testing for current in a cable with five color-coded wires, the author used a meter to test two wires at a time. How many different tests are required for every possible pairing of two wires? 15. Jumble Many newspapers carry “Jumble,” a puzzle in which the reader must unscramble letters to form words. The letters MHRHTY were included in newspapers on the day this exercise was written. How many ways can those letters be arranged? Identify the correct unscrambling, then determine the probability of getting that result by randomly selecting one arrangement of the given letters. 16. DNA Nucleotides DNA (deoxyribonucleic acid) is made of nucleotides. Each nucleotide can contain any one of these nitrogenous bases: A (adenine), G (guanine), C (cytosine), T (thymine). If one of those four bases (A, G, C, T) must be selected three times to form a linear triplet, how many different triplets are possible? All four bases can be selected for each of the three components of the triplet. 17. Powerball As of this writing, the Powerball lottery is run in 44 states. Winning the jackpot requires that you select the correct five different numbers between 1 and 69 and, in a separate drawing, you must also select the correct single number between 1 and 26. Find the probability of winning the jackpot. 18. Teed Off When four golfers are about to begin a game, they often toss a tee to randomly select the order in which they tee off. What is the probability that they tee off in alphabetical order by last name? 19. ZIP Code If you randomly select five digits, each between 0 and 9, with repetition allowed, what is the probability you will get the author’s ZIP code? 20. Age Discrimination The Cytertonics Communications Company reduced its management staff from 15 managers to 10. The company claimed that five managers were randomly selected for job termination. However, the five managers chosen are the five oldest managers among the 15 that were employed. Find the probability that when five managers are randomly selected from a group of 15, the five oldest are selected. Is that probability low enough to charge that instead of using random selection, the company actually fired the oldest employees? 21. Phone Numbers Current rules for telephone area codes allow the use of digits 2–9 for the first digit, and 0–9 for the second and third digits, but the last two digits cannot both be 1 (to avoid confusion with area codes such as 911). How many different area codes are possible with these rules? That same rule applies to the exchange numbers, which are the three digits immediately preceding the last four digits of a phone number. Given both of those rules, how many 10-digit phone numbers are possible? Given that these rules apply to the United States and Canada and a few islands, are there enough possible phone numbers? (Assume that the combined population is about 400,000,000.) 22. One Mississippi The counting sequence of “one Mississippi, two Mississippi, three Mississippi, . . .” is often used because saying a number with “Mississippi” takes about one second. A classic counting problem is to determine the number of different ways that the letters of “Mississippi” can be arranged. Find that number. If the letters are mixed up in a random sequence, what is the probability that the letters will be in alphabetical order?
RkJQdWJsaXNoZXIy NjM5ODQ=