Consider the design of a communication system. (1) How many three-digit phone prefixes that are used to represent a particular geographic area

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?

(1) We have to form three-digit phone prefixes. For each we have 10 choices. So in total we have 10 x 10 x 10 = 1000 numbers.

(2) Number should not start with 0 or 1, but should contain 0 or 1 as the middle digit.

For first place we have 8 choices ( 2 - 8).

For second place we have 2 choices (0,1), while for third we have 10 choices.

So, 8 x 2 x 10 = 160 numbers are possible.

(3) No digit should appear more than once in each prefix.

For first place we have 10 choices while 9 for second, and 8 for third as no number can repeat. 28 such total numbers are 10 x 9 x 8 = 720.