Number of Possible Telephone Numbers

Date: 7/15/96 at 21:23:43
From: Anonymous
Subject: Number of Possible Telephone Numbers

How many phone numbers can be made under the following conditions:
(First digit cannot be 0 or 1 because you'll get the operator or long
distance.)
a) The first two digits are 3 followed by 6
b) The third digit is even 
c) The fourth digit is greater than 5
d) The fifth and seventh digits are odd
e) The sixth digit is 2

Date: 7/15/96 at 22:1:32
From: Doctor Sydney
Subject: Re: Number of Possible Telephone Numbers

Hello!

This is a great question, and it is from an area of math called 
combinatorics, which is a fancy way to say "counting."  There are 
people who make their livings thinking about counting problems!  

One way to approach problems like these is to count and see how many 
choices we have for each digit and then multiply them together.  
Assuming that you are talking about 7 digit phone numbers, we note the 
following:

1) For the first digit, we have only one choice -- we must choose a 3.
2) For the second digit, we have only one choice -- we must choose a 	
   6.
3) For the third digit, we have five choices -- we can choose 0, 2, 4, 
   6, or 8.
4) For the fourth digit, we have four choices -- we can choose 6, 7, 
   8, or 9.
5) For the fifth digit, we have five choices -- we can choose 1, 3, 5, 
   7, or 9.
6) For the sixth digit, we have one choice -- we must choose 2.
7) For the seventh digit, we have five choices -- we can choose 1, 3, 
   5, 7, or 9.

Okay, that said and done, we can solve the problem.  Actually, why 
don't we solve an easier problem, and then armed with the techniques 
for solving that similar problem and given the above information, you 
should be able to solve this problem on your own.  

Okay, so let's think about this problem:  
How many different license plates can be made with only two "digits" 
where the first "digit" is a letter, and the second "digit" is a 
number?  So, license plates that look like the following qualify: G9, 
U2, A1, etc...

License plates not qualifying include the following: 9G (Here the 
number is first whereas the letter should really be first), 88 (no 
letter), rt355 (too long!).  Okay, I think you have the idea.  

So, let's again look at the choices we have:

1) For the first "digit" we have 26 possibilities -- the 26 letters of 
   the alphabet.

2) For the second "digit" we have 10 possibilities -- 0, 1, 2, ...,9.

Let's do the counting in several stages:

Suppose the first digit is an "A."  Since there are 10 possibilities 
for the second digit, we have 10 possibilities for license plates 
beginning with the letter "A."  Similarly, for every other letter, 
there are 10 possibilities for license plates beginning with that 
letter.  Since there are 26 letters, and for each letter we have 10 
possible license plates, there are a total of 26*10 = 260 possible 
license plates.

Okay, so one problem is solved.  But, you say, THAT wasn't MY 
question.  Well, it wasn't, but I hope that it will help you to answer 
your question.  Using strategies from the second problem, and the 
setup I gave you for your problem, why don't you see if you can solve 
your problem.  If you want to check your answer and reasoning, please 
write back!  

Hope this helps, and good luck.  You can tell all of your friends at 
school that you are a combinatorist (that is a fancy way for saying 
someone who does these counting problems!).

-Doctor Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 7/16/96 at 21:54:27
From: Anonymous
Subject: Dr.Sydney (I Solved it!)

Thanks for helping me with that question, my answer is 500 phone
numbers. I'm sure it's right. I got it by doing this:

1 X 1 X 5 X 4 X 5 X 5 = 500 possible numbers.

Connie

Date: 7/17/96 at 15:2:57
From: Doctor Sydney
Subject: Re: Dr.Sydney (I Solved it!)

Dear Connie,

Great job!  You did it perfectly.  I hope you continue to find fun 
math problems to work on, and feel free to write us again if you have 
any questions or things you are curious about!

-Doctor Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/