Question Video: Applications of the Counting Principle That Involve Replacement

Phone numbers on a particular network are twelve-digit long, where the first three digits are always 072. Calculate the total number of different phone numbers which the network can use.

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Video Transcript

Phone numbers on a particular network are 12 digits long, where the first three digits are always zero seven two. Calculate the total number of different phone numbers which the network can use.

We’re told that the phone numbers are all 12 digits long, but that the first digits are always zero seven two. This means what we’re really being asked is how many different numbers can we make using the remaining nine digits. And of course, we know that in telephone numbers the digits can be repeated. So, let’s consider this in turn.

We can use the digits zero through nine in our phone number. That gives us a total of 10 options for each of our digits. And so, it follows that there are 10 ways that we can choose the first digit of the phone number. Since these numbers can be repeated, we know that there are also 10 ways to choose the second digit. We can choose any number from zero through to nine. There are 10 ways to choose the third digit, and so on. Each time we pick a new digit, there’s 10 numbers that we can pick. So, what do we do with all of these numbers?

Well, we recall something called the counting principle or the product rule for counting. This says that to find the total number of outcomes for two or more events, we multiply the number of outcomes for each event together. And so, we see the total number of different phone numbers the network can use is found by multiplying each of these 10s together or, indeed, by calculating 10 to the ninth power. 10 to the ninth power is one billion, meaning that there are one billion different phone numbers that the network can use.

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