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The number of arrangements when order matters

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Wrong Number

All phone numbers are unique, and dialing the wrong number will connect you to the wrong person. Writing down the wrong number or programming the wrong number into your cell phone could be the difference between getting a date, making a job connection, getting a ride home?or not. The order of the numbers in a phone number matter, but how do we decide which unique ordering to assign to a phone?

Why It Matters

Phone numbers are an example of permutations. In the United States, each phone number consists of ten digits, three for the area code and seven for the unique phone number. (There is also a ?1? preceding each phone number to indicate that the first three digits are the area code.) So, how many possible phone numbers are there? Well, this could turn out to be a simple permutation problem: you have ten number choices (0-9) for each digit of a phone number and repetitions are allowed. Technically, there could be as many as \begin{align*}10^{10} = 10,000,000,000\end{align*}, or 10 billion possible phone numbers in the U.S.

However, there are certain restrictions placed on phone numbers that we haven?t taken into account. An area code or phone number cannot start with a ?0? because dialing zero is traditionally reserved for connecting with the telephone operator. An area code cannot start with a ?1? because that is reserved to indicate that an area code is about to be dialed. There are also a few specific area codes that are off limits: 911 is reserved for emergency calls, 611 is reserved for the telephone company, and 411 is reserved for information or local directory assistance. Given these restrictions, we need to eliminate all of these unusable possibilities from our original total of 10,000,000,000.

Numbers beginning with 0: \begin{align*}10^9 = 1,000,000,000\end{align*}

Numbers beginning with 1: \begin{align*}10^9 = 1,000,000,000\end{align*}

Numbers beginning with 911: \begin{align*}10^7 = 10,000,000\end{align*}

Numbers beginning with 411: \begin{align*}10^7 = 10,000,000\end{align*}

Numbers beginning with 611: \begin{align*}10^7 = 10,000,000\end{align*}

Total unusable numbers: 2,030,000,000

Therefore, the total number of possible phone numbers is 7,970,000,000.

As you can imagine, with a national population greater than 300 million, and with many people possessing more than one phone number, there may come a time very soon when phone numbers in the U.S. will need to be lengthened to 8 digits to accommodate the need for more unique numbers.

See for yourself: http://www.artlebedev.com/mandership/91/

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If the United States had to switch to 8-digit phone numbers, how many possible numbers would be available for use?

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