If using numbers 0 through 9 how many possiblites are there for a seven digit phone number?

Answer 1

Assuming you could literally put any number into any spot (meaning you can start with zero or do 911-XXXX), then you have 10 options for each of 7 spaces, so it's 10^7=10,000,000 (ten million) possibilities.

But numbers can't start any way they choose in real life. You can't do zero first, since that's reserved to call the operator. So we have nine options first, then 6 of 10, so it's 9*10^6 = 9 million.

Now scratch out 411-xxxx and 911-xxxx since hitting those first three digits brings up information or emergency response, respectively. So we've taken out 10^4 (10,000) twice, leaving (9*10^6) - (2*10^4) = 8,980,000.

Oh, and don't forget 555-xxxx, which is an intentionally dead prefix that can be used in movies, TV, books, etc. so that they aren't giving a person's real number. Except that's not entirely true...555-1212 connects to a directory assistant in the US, and 1-800-555-1111 connects to Bell Canada. In fact, only 555-0100 through 555-0199 are reserved for fictitious use now; any others may be put to use someday. So that's 100 more numbers we can remove, bringin us down to

(9*10^6) - (2*10^4) - 100 = 8,979,900 genuinely possible 7-digit phone numbers

out of

10,000,000 theoretically possible 7-digit phone numbers.