Counting Up Three Digit Numbers

Date: 10/27/98 at 11:49:47
From: John Harrigan
Subject: Math

How many possible 3-digit numbers are there that have a middle zero?
How many possible 4-digit numbers are there that have two middle zeros?

Date: 10/31/98 at 00:55:00
From: Doctor Teeple
Subject: Re: Math

Hi John,

Those are good questions. 

What if we look at the first question in a different light? What if I 
say "How many possible 3-digit numbers are there that have a middle 
zero AND begin with one?" We can list those. I count ten:

   100, 101, 102, 103, 104, 105, 106, 107, 108, and 109

Now, what if we change all of the beginning 1's to 2's? We'd have the 
list of 3-digit numbers that have a middle zero AND begin with 2. So 
there are 10 of those. 

We could continue to do the same thing, replacing the 2 with 3, 4, 5, 
6, 7, 8, and 9, counting each of those individual lists. Do you see how 
we would cover all of the possible 3-digit numbers that have a middle 
zero? If we count up each of the individual lists that we make, we will 
get all of the possible 3-digit numbers that have a middle zero.

But notice that the lists are the same, except for the first digit. 
This means that we don't have to write down all of the numbers for each 
different first digit. Once we know there are 10 numbers in the list of 
3-digit numbers that have a middle zero and begin with one, we know 
there are 10 in each of our other lists. So all we have to do is add 10 
to the total for each list we could make. How many times do we add 10 
to the total? 

The second problem is very similar to the first one. You can answer it 
the same way we answered the first. Please write back if you need help 
with that problem or need more help with the problem that we did.

- Doctor Teeple, The Math Forum
  http://mathforum.org/dr.math/