combinatorics (number of phone numbers)

[xc630]

Hello I am having a lot of trouble with this one problem:

Telephone nubmers in the U.S. and Canada have 10 digits as follows.

3 digit area code number: 1st digit is not 0 or 1
. . . . . . . . . . . . . . . . . . . . .2nd digit must be 0 or 1
3 digit exchange number: 1st and 2nd digits are not 0 or 1
4 digit line number:. . .. . . not all zeros

Suppose a state has a 5 telephone area codes. Tell how many phone numbers there could be in the state without adding any more area codes.

Find how many 7 digit phone numbers are possible in one area code and then multiply by five to get the answer.

I tried and got it woring and the answer is supposedly 30 million something, but I do not know where I went worng.

I tried this:

8x8x10x9x10x10x10

...following the rules for 3digit exchange number and 4 digit line number and got 5,760,000. I then multiplied this by five and got only 28,800,000.

In the back the answer says I should have gotten 6399360 1st and multiplied by five and got 31996800. I would aprreciate any help in identifying where I went wrong. Thank you.

 

[pka]

Re: combinatorics

a state has a 5 telephone area codes. Tell how many phone numbers there could be in the state without adding any more area codes.
I was told to find how many 7 digit phone numbers are possible in one area code and then multiply by five to get the answer.
In the back the answer says I should have gotten 6399360 1st and multiplied by five and got 31996800.

Now \(\displaystyle \L
8 \cdot 8 \cdot 10 \cdot \underbrace { - - - - }_{10^4 - 1} = {\rm{6399360}}\) why do we subtract 1 from 10<SUP>4</SUP>?

 

[xc630]

I am not sure. I sit because there are 10000 4 digit numbers and minus one would mean the one that is not all zeros?

 

[pka]

there are 10000 4 digit numbers and minus one would mean the one that is not all zeros?

You have it!
Only one is all zeros: '0000'.
All others have at least one non-zero.