Phone Numbers Current rules for telephone area codes allow the...

Question

Phone Numbers Current rules for telephone area codes allow the use of digits 2–9 for the first digit, and 0–9 for the second and third digits, but the last two digits cannot both be 1 (to avoid confusion with area codes such as 911). How many different area codes are possible with these rules? That same rule applies to the exchange numbers, which are the three digits immediately preceding the last four digits of a phone number. Given both of those rules, how many 10-digit phone numbers are possible? Given that these rules apply to the United States and Canada and a few islands, are there enough possible phone numbers? (Assume that the combined population is about 400,000,000.)

Accepted Answer

Welcome back everyone to another video. A security system uses a 6 digit passcode with the following rules. The first digit must be between 3 and 7 inclusive. The next two digits can be any number from 0 to 9. The last three digits can also be any number from 0 to 9, except that all three digits cannot be the same. To avoid passcodes like 111. 222, etc. Number 1, how many different six digit passcodes are possible under these rules? Number 2, given that the system is used by 150 million devices, are there enough available passcodes? Let's begin solving this problem with the first part. We want to form a six digit passcode. So what we're going to do is basically draw 6 boxes 12345, and 6 and we're going to think about these rules. So it says that the first digit must be between 3 and 7 inclusive. How many choices do we have for the first digit? Well, that would be 7 minus 3. The higher value minus the lower value plus 1. This is how we calculate the total number of elements between 3 and 7 inclusive, and we get 5 total, so +3456, and 7, right? So we have 5 choices for the first box. The next two digits can be any number from 0 to 9. Once again, we are going to calculate the number of available elements, 9 minus 0 + 1. We have 10 possible digits to fill the next two boxes. So 10 and 10. Now it says that the last three digits can also be any number from 0 to 9, except that all three digits cannot be the same. So what we're going to do is basically understand that. The last three digits. Can be from 0 to 9, so we have a total of 10 digits available. And the total number of ways to fill these 3 boxes would be again 10. Some and some, but we have to subtract. The ways which produce three exact same digits, so those would be 111,000222, and so on. And those would be a total of 10 ways, right? 000111, and so on up to 999. So there's a total of 10. Ways to get three exactly same digits. So what we're going to do is use the multiplication rule to identify the total number of arrangements. The multiplication rule says that we take 5, we multiply by 10. Followed by multiplication by 10, and this is where we have to be careful. First of all, we're going to tweet the remaining 3 boxes. Yes A separate part of the code, right? So what we're going to do is basically continue with the multiplication rule, but now we are going to think about the total number of ways to form a 3 digit code from 0 to 9. That would be 10, multiplied by 10, multiplied by 10. But we have to subtract those 10. Ways which produce the exact same digits, so we take 10 cubed minus 10. And this is our solution to part number one, so we can basically simplify it and say that we are multiplying 5 by 102, which is multiplied by 10 cubed minus 10. And if we perform the calculation, we're going to get 495,000. This is the total number of six digit passcodes. And now for number 2, We have a total of 150 million devices, right? If we wanted to assign a unique code for each device, there would be not sufficient codes because we only have 495,000 available, meaning for number 2, we can basically say that There is. An insufficient number. Sufficient Number Of available. Passcos Such that each device gets a unique passcode, right? And that would be our answer to the second part of the problem. That would be it, and thank you for watching.

Key Concepts

Combinatorial Counting

Permutations and Combinations

Population vs. Phone Number Capacity

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