Answer to Question #117113 in Combinatorics | Number Theory for Priya

Question

In a certain country telephone numbers have 9 digits. The first two digits are the area code (03) and are the same within a given area. The last 7 digits are the local number and cannot begin with 0. How many different telephone numbers are possible within a given area code in this country?

Answer 1

The telephone numbers in a certain country have 9 digits. The first 2 digits are the area code (03) and are the same within a given area.

Since, there is no restriction of repetition of the numbers, the remaining 7 digits of the local number can be filled up using 10 different digits 0, 1, 2, ... , 9.

Therefore, the total number of arrangements = 10⁷.

But, the local number cannot begin with 0.

To find the numbers beginning with 0, we fix 0 at the beginning of the 7 digit number and the remaining 7 - 1 = 6 digits can be filled up using 10 different digits.

i.e. the number of 7 digit numbers that begin with 0 = 10⁶.

Hence, the total number of 7 digit numbers that do not begin with 0 = 10⁷ - 10⁶ = 9000000.

Since, the first 2 digits are same within a given area

Therefore, the number of different telephone numbers that are possible within a given area code in this country is 9000000 (Ans).