a) Consider the design of a communication system.
i. How many three-digit phone prefixes that are used to represent a particular geographic area (such as an area code) can be created from the digits 0 through 9?
ii. As in part i., how many three-digit phone prefixes are possible that do not start with 0 or 1, but contain 0 or 1 as the middle digit?
iii. How many three-digit phone prefixes are possible in which no digit appears more than once in each prefix?
i. To find the number of three-digit phone prefixes that can be created from the digits 0 through 9, we need to consider that the first digit cannot be zero, as phone numbers usually don't start with zero. Therefore, we have 9 choices for the first digit (from 1 to 9).
For the second and third digits, we have 10 choices for each digit (from 0 to 9).
So, the total number of three-digit phone prefixes that can be created is:
9 choices for the first digit * 10 choices for the second digit * 10 choices for the third digit = 900 prefixes.
ii. The question states that the three-digit phone prefixes cannot start with 0 or 1, but they can contain 0 or 1 as the middle digit.
For the first digit, we have 8 choices (from 2 to 9) since we cannot use 0 or 1.
For the second digit, we have 2 choices (0 or 1) since it must be either 0 or 1.
For the third digit, we have 10 choices (from 0 to 9).
So, the total number of three-digit phone prefixes that satisfy these conditions is:
8 choices for the first digit * 2 choices for the second digit * 10 choices for the third digit = 160 prefixes.
iii. To calculate the number of three-digit phone prefixes where no digit appears more than once in each prefix, we need to consider that we have the same restrictions as in part ii (no 0 or 1 as the first digit, and 0 or 1 as the middle digit is allowed).
For the first digit, we have 8 choices (from 2 to 9).
For the second digit, excluding the first digit chosen, we have 9 choices (from 0 to 9 except for the first digit chosen).
For the third digit, we have 8 choices (from 0 to 9 except for the two digits chosen previously).
So, the total number of three-digit phone prefixes where no digit appears more than once is:
8 choices for the first digit * 9 choices for the second digit * 8 choices for the third digit = 576 prefixes.