Answer to Question #148096 in Combinatorics | Number Theory for Talena

Question

In a certain country, telephone numbers have 9 digits. The first two digits are the area code and are the same within a given area. The last seven digits are the local number and can’t begin with 0. How many different numbers are possible within a given area code in this country?

Answer 1

Since it is within a given area code, that implies the first two digits are the same

[ 1 ] × [ 1 ] = 1 way. There's only one way of choosing the area code. That implies the area code is not subject to change.

The remaining seven digits can come from any 10 digits between 0-9.

Since the first digit can't be zero, that implies the first digit can only be chosen from any 9 digits between 1-9. The 2nd digit can be chosen from any 10 digits 0-9. Ditto for the 3rd up to the 7th digit.

We have

[9] × [10] × [10] × [10] × [10] × [10] × [10] = 9 000 000 ways. There are 9 000 000 ways to choose the local numbers.

The number of possible different numbers within a given area code is 1 × 9 000 000 = 9 000 000 ways