# How many different phone numbers are possible within an area code?

A phone number is composed of 10 digits. The first three are the area code the other 7 are the local telephone number which cannot begin with a 0. How many different telephone numbers are possible in a single area code?

• "Integers" is the wrong word here. It's ten digits. For example $42$ is an integer but it is not a digit; rather it is expressed with two digits. – Michael Hardy Sep 6 '14 at 1:49
• The area code does not matter in this problem. Now, for the remaining $7$ digits of the number, first cannot be $0$, so we have $9$ choice for it, for rest of the $6$ digits we have $10$ choices, multiplying all we get $9\times 10^6$ – Vikram Sep 6 '14 at 4:49

If we visualize the phone number as having three "slots": one for the area code, one for the first digit and the last being the six remaining digits, by the multiplication principle, there is 1 way to complete filling the first slot, 9 ways to complete the second slot, and $10*10*10*10*10*10$ ways to fill the last slot; giving $1*9*10^6$ ways.

From Wikipedia http://en.wikipedia.org/wiki/North_American_Numbering_Plan:

Each three-digit area code may contain up to 7,919,900 unique phone numbers:

• NXX may begin only with the digits [2–9], providing a base of 8 million numbers: ( 8 x 100 x 10000 ) .
• However, the last two digits of NXX cannot both be 1, to avoid confusion with the N11 codes (subtract 80,000).
• Despite the widespread usage of NXX "555" for fictional telephone numbers — see 555 (telephone number) — today, the only such numbers specifically reserved for fictional use are "555-0100" through "555-0199", with the remaining "555" numbers released for actual assignment as information numbers (subtract 100).
• In individual geographic area codes, several other NXX prefixes are generally not assigned: the home area code(s), adjacent domestic area codes and overlays, area codes reserved for future relief nearby, industry testing codes (generally NXX 958 and 959) and special service codes (such as NXX 950 and 976). Subtract for 911 411etc emergency and informational numbers
• This should be the accepted answer. – MarcusJ Apr 10 '15 at 23:53

well it is going to be 10,000,000 (000-0000 to 999-9999) minus any number before 100-0000 that would make 9,000,000 legal numbers.