<s>[INST] This problem requires a review of the material covered in this chapter along with an understanding of counting principles discussed in previous chapters. In a particular city, two companies, namely, \(A\) and \(B\), provide bus transportation services to the citizens. To measure the quality of their services, the city officials plan to conduct a survey to gather data on the number of passengers each company carries daily. For the first day of the survey, it has been decided to select one bus from each company and record the number of passengers it carried during the day. The buses selected for the survey had the following numbers: company \(A\) (206, 742), and company \(B\) (123, 872). The company officials, however, are interested in knowing the total number of different four-digit numbers that can be formed using the digits of the given two numbers, with the condition that the first digit cannot be zero. Solution: _To obtain the answer, we need to recognize that the four-digit numbers to be formed are a combination of digits of numbers 206 and 742 from company \(A\) and digits of numbers 123 and 872 from company \(B\). That is, we are taking digits from the set 0, 1, 2, 3, 6, 7, 8, 9, 2. However, we should not repeat any digit more than the number of times it appears in the given numbers; that is, digit 2 appears twice. Because the first digit of any number formed cannot be zero, we have two cases to consider. Case 1: The first digit is 0. In this case, there are eight digits (1, 2, 2, 3, 6, 7, 8, 9) from which to choose the remaining three digits. Because digit 2 appears twice, we need to take into account its repeated appearances. Thus, for the second digit, we have eight choices--any of the eight digits including the second 2. For the third digit, we have seven choices, and for the fourth digit, six choices. Hence, the total number of different numbers we can form is 8 \(\times\) 7 \(\times\) 6 = 336. Case 2: The first digit is not 0. In this case, the first digit can be any of the digits 1, 2, 2, 3, 6, 7, 8, 9. Although there are eight digits, two of them are the same digit 2. Hence, for the first digit, we have eight choices. For the second digit, there are seven choices, and for the third digit, six choices, and so on. Hence, the total number of different numbers we can form is 8 \(\times\) 7 \(\times\) 6 = 336. Thus, the total number of different numbers that can be formed is 336 + 336 = 672._[/INST]