Question

How many different 7-digit phone numbers can be used by a phone company?

The phone numbers cannot start with a zero.

Answer 1

numbers from 1000000 to 9999999

Answer 2

Okay, how many possibilities do we have for the first digit? It can be any digit other than \(0\), so we have \(9\) possibilities. How many possibilities for the second? Any of the \(10\) digits. How many for the third digit? \(10\) as well. There's a neat combinatorial principle called the rule of product which lets us count the total number of phone numbers by multiplying the number possibilities for each digit: \(9\times10\times10\times\cdots=9\times10^6=9000000\). This is an extension of the very sample principle at work when we count the number of *combinations* of, say, three pairs of jeans with two shirts: \(3\times2=6\).

Answer 3

@Jello_Submarine provides an alternate but also valid solution. The total number of integers starting with \(1000000\) up to and including \(9999999\) = \(9999999-1000000+1=9000000\).

Answer 4

oops, where I wrote sample I meant simple.

Answer 5

Ah Thankyou! this is what I get for not taking a break! :p