The Telephone Numbering Plan in North America: Counting Possible Numbers

[bunyonb]

The Telephone Numbering Plan The North American numbering plan (NANP) specifies the format of telephone numbers in the U.S., Canada, and many other parts of North America. A telephone number in this plan consists of 10 digits, which are split into a three-digit area code, a three-digit office code, and a four-digit station code. Because of signaling considerations, there are certain restrictions on some of these digits. To specify the allowable format, let X denote a digit that can take any of the values 0 through 9, let N denote a digit that can take any of the values 2 through 9, and let Y denote a digit that must be a 0 or a 1. Two numbering plans, which will be called the old plan, and the new plan, will be discussed. (The old plan, in use in the 1960s, has been replaced by the new plan, but the recent rapid growth in demand for new numbers for mobile phones and devices will eventually make even this new plan obsolete. In this example, the letters used to represent digits follow the conventions of the North American Numbering Plan.) As will be shown, the new plan allows the use of more numbers. In the old plan, the formats of the area code, office code, and station code are NYX, NNX, and XXXX, respectively, so that telephone numbers had the form NYX-NNX-XXXX. In the new plan, the formats of these codes are NXX, NXX, and XXXX, respectively, so that telephone numbers have the form
NXX-NXX-XXXX.

How many different North American telephone numbers are possible under the old plan and under the new plan?

MY ATTEMPT at answering this question:

It is said that there are two ways to count the product rule and the sum rule.

The product rule says that procedure can be broken down into a sequence of two
tasks. If there are {n}^{1} ways to do the first task, and for each of these ways of doing the first task, there are {n}^{2} ways to do the second task, then there are {n}^{1}{n}^{2} ways to do the procedure.

Thus, by the product rule there are 8*2*10=160 area codes with format NYX and 8*10*10=800 area codes with format NXX. Similarly, by the product rule, there are 8*8*10=640 office codes with format NNX. The product rule also shows that there are 10*10*10*10. = 10,000 station codes with format XXXX. Consequently, applying the product rule again, it follows that under the old plan there are 160*640*10,000=1,024,000,000. Different numbers available in North America. Under the new plan, there are 800*800*10,000=6,400,000,000 different numbers available. Am I on the right track?

 

[MarkFL]

Your application of the product rule is correct, and so just to check your math:

Old plan:

\(\displaystyle N_O=2\cdot8^3\cdot10^6=1,024,000,000\quad\checkmark\)

New plan:

\(\displaystyle N_N=8^2\cdot10^8=6,400,000,000\quad\checkmark\)

 

[bunyonb]

Your application of the product rule is correct, and so just to check your math:

Old plan:

\(\displaystyle N_O=2\cdot8^3\cdot10^6=1,024,000,000\quad\checkmark\)

New plan:

\(\displaystyle N_N=8^2\cdot10^8=6,400,000,000\quad\checkmark\)

Thank you. For some reason the Latex format doesn't render for me.

 

[MarkFL]

Thank you. For some reason the Latex format doesn't render for me.

You have to wrap your code in tags, the easiest of which to use are the [MATH][/MATH] tags, which can be generated by using the $\displaystyle \sum$ button on the toolbar. :)

 

[bunyonb]

You have to wrap your code in tags, the easiest of which to use are the [MATH][/MATH] tags, which can be generated by using the $\displaystyle \sum$ button on the toolbar. :)

Ah ok thank you for this information.