4-4 Counting 181 Multiplication Counting Rule: Hacker Guessing a Password EXAMPLE 1 A computer hacker finds that a password is entered as •••••, so the characters are hidden, but we can see that there are five characters. We can use 92 different characters with a typical keyboard. How many different passwords are possible using five characters? If the hacker starts to generate all different possibilities, what is the probability of guessing the correct password on the first attempt? YOUR TURN. Do Exercise 5 “Pin Numbers.” SOLUTION There are 92 different possibilities for each character, so the total number of different possible passwords is n1 # n2 # n3 # n4 # n5 = 92# 92# 92# 92# 92 = 6,590,815,232. Because all of the passcodes are equally likely, the probability of getting the correct passcode on the first attempt is 1>6,590,815,232 or 0.000000000152. Unfortunately, there are sophisticated programs that allow hackers to try large numbers of likely possibilities, beginning with the commonly used but highly ineffective password of “password.” 2. Factorial Rule The factorial rule is used to find the total number of ways that n different items can be rearranged (order of items matters). The factorial rule uses the following notation. FACTORIAL RULE The number of different arrangements (order matters) of n different items when all n of them are selected is n!. NOTATION The factorial symbol (!) denotes the product of decreasing positive whole numbers. For example, 4! = 4# 3# 2# 1 = 24. By special definition, 0! = 1. The factorial rule is based on the principle that the first item may be selected n different ways, the second item may be selected n - 1 ways, and so on. This rule is really the multiplication counting rule modified for the elimination of one item on each selection. Factorial Rule: Scrambling Letters EXAMPLE 2 Some words consist of letters which can be rearranged to form other words. Consider the word “steam.” a. How many different ways can the letters of “steam” be arranged? b. If the letters of “steam” are arranged randomly, what is the probability that the letters will be in alphabetical order? c. How many different words can be formed with the letters in “steam”? continued Changes to Mega Millions Lottery The old Mega Millions lottery cost $1 and a player would select 5 numbers from 1 to 75 and a Mega number from 1 to 15; there is 1 chance in 258,890,850 of winning the jackpot. With the new format introduced on October 28, 2017, a ticket costs $2 and a player selects 5 numbers from 1 to 70 and a Mega number from 1 to 25; there is 1 chance in 302,575,350 of winning the jackpot. The change was motivated by psychological factors based on reasoning that with a much lower chance of winning the jackpot, the jackpot would grow to much larger amounts. Larger jackpots generate more lottery ticket sales, resulting in greater income from the lottery. In October of 2018, one player won the jackpot of $1.6 billion. The winner taking the lump sum payout would get about $904 million before taxes, and about $500 million after taxes.
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