184 CHAPTER 4 Probability Permutations Rule (with Some Identical Items): Good Survey Design EXAMPLE 4 When designing surveys, pollsters often repeat a question to see if a subject is thoughtlessly providing answers just to finish quickly. For one survey with 10 questions, 2 of the questions are identical to each other (with superficial differences in wording), and 3 other questions are also identical to each other (with superficial differences in wording). For this survey, how many different arrangements are possible? Is it practical to survey enough subjects so that every different possible arrangement is used? YOUR TURN. Do Exercise 9 “Statistics Counts.” SOLUTION We have 10 questions with 2 that are identical to each other and 3 others that are also identical to each other, and we want the number of permutations. Using the rule for permutations with some items identical to others, we get n! n1!n2! cnk! = 10! 2!3! = 3,628,800 2# 6 = 302,400 INTERPRETATION There are 302,400 different possible arrangements of the 10 questions. It is not practical to accommodate every possible permutation. For typical surveys, the number of respondents is somewhere around 1000. 5. Combinations Rule The combinations rule is used when there are n different items available for selection, only r of them are selected without replacement, and order does not matter. The result is the total number of combinations that are possible. (Remember: Rearrangements of the same items are considered to be the same combination.) COMBINATIONS RULE When n different items are available, but only r of them are selected without replacement, the number of different combinations (order does not matter) is found as follows: nCr = n! 1n - r2!r! Combinations Rule: Lottery EXAMPLE 5 In Florida’s Cash 5 lottery game, winning the jackpot requires that you select 5 different numbers from 1 to 35, and the same 5 numbers must be drawn in the lottery. The winning numbers can be drawn in any order, so order does not make a difference. a. How many different lottery tickets are possible? b. Find the probability of winning the jackpot when one ticket is purchased. Choosing Personal Security Codes All of us use personal security codes for ATM machines, Internet accounts, and home security systems. The safety of such codes depends on the large number of different possibilities, but hackers now have sophisticated tools that can largely overcome that obstacle. Researchers found that by using variations of the user’s first and last names along with 1800 other first names, they could identify 10% to 20% of the passwords on typical computer systems. When choosing a password, do not use a variation of any name, a word found in a dictionary, a password shorter than seven characters, telephone numbers, or social security numbers. Do include nonalphabetic characters, such as digits or punctuation marks. A p r A I c h T o is th You Can Be Very Special! If you take a deck of 52 cards and thoroughly shuffle it, it is extremely unlikely that the order of the cards has occurred in the past and it is extremely unlikely that this order will occur again in the future. Don’t believe that? Well, there are 52! possible arrangements of the deck, where 52! = 8.07 * 1067 (or the 68-digit number 80,658,175, . . .). If every living human could shuffle a deck every five seconds, the number of centuries it would take all living humans to do that many shuffles would be a number that is 50 digits long. That is the power of factorials!
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