# Discrete math problems

1. A telephone number is a ten-digit number whose first digit cannot be a 0 or a 1. How many telephone numbers are possible?

2. A computer operating system allows files to be named using any combination of uppercase letters (A-Z) and digits (0-9), but the number of characters in the file name is at most eight (and there has to be at least one character in the file name). For example, X23, W, 4AA and ABCD1234 are valid file names, but W-23 and WONDERFUL are not valid (the first has an improper character, the second is too long). How many different file names are possible in this system?

3. A class contains ten boys and ten girls. In how many different ways can they stand in a line if they must alternate in gender (no two boys and no two girls are standing next to eachother)?

#### Solution Preview

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1. First digit can be filled using the digits 2, 3, 4, 5, 6, 7, 8 or 9. Hence the first digit of the telephone number can be filled in 8 different ways.

Each of the remaining digits can be filled using 0, 1, ..., 9. Hence each of the remaining digits can be filled in 10 different ways.

Hence the number of possible telephone numbers = 8 * (10)9

= 8000000000

2. Given that the file name has at least one character and at most eight characters. There are 26 letters and 10 digits are ...

#### Solution Summary

The discrete math problems for computer operation systems are discussed.