Counting Even Digits in Three-Digit Numbers

Date: 10/24/2004 at 11:27:30
From: Kevin
Subject: 3 digit even number problem.

How many 3-digit numbers are there in which the number of even digits
is even?

I am having trouble finding a way to solve it. I started trying to
find the number of combinations of even numbers, but it isn't working
very well.

Date: 10/24/2004 at 12:53:59
From: Doctor Greenie
Subject: Re: 3 digit even number problem.

Hi, Kevin.

This is an interesting problem which can be tackled using any of 
several different approaches.  One very good approach is to look at 
the numbers of combinations of even digits, as you seem to suggest.

We want the number of even digits to be even, so we can have either 0 
or 2 even digits in our 3-digit number.  If we represent even digits 
with E and odd digits with O, then we can have 3-digit numbers of any 
of these forms:

  EEO
  EOE
  OEE
  OOO

We can find the numbers of different 3-digit numbers of each of these 
forms using the fundamental counting principle.  In doing this, we 
need to remember that we have 5 choices for each even or odd digit, 
except in the case of a leading even digit, where we have only 4 
choices, since a leading digit of 0 is not allowed.  So we have

  EEO:  4*5*5 = 100 3-digit numbers of this form
  EOE:  4*5*5 = 100 3-digit numbers of this form
  OEE:  5*5*5 = 125 3-digit numbers of this form
  OOO:  5*5*5 = 125 3-digit numbers of this form
  -----------------------------------------------
        total:  450

So there are 450 3-digit numbers with an even number of even digits.

To give us some confidence in our method, we can use the same process 
to find the number of 3-digit numbers with an odd number of even 
digits.  Together with the number of 3-digit numbers with an even 
number of even digits, this should give us a grand total of 900, which 
is the total number of 3-digit numbers.

  EEE:  4*5*5 = 100 3-digit numbers of this form
  EOO:  4*5*5 = 100 3-digit numbers of this form
  OEO:  5*5*5 = 125 3-digit numbers of this form
  OOE:  5*5*5 = 125 3-digit numbers of this form

This total is also 450, giving us the required grand total of 900 3-
digit numbers, so our method and our computations are probably 
correct.

Note that the numbers of 3-digit numbers with an even number of even 
digits and with an odd number of even digits are the same--450 each.  
There is a very sophisticated method for solving this problem very 
quickly using this fact; we show that exactly half of all 3-digit 
numbers have an even number of even digits, so the number of 3-digit 
numbers with an even number of even digits is half of 900, or 450.

Following is an informal argument for why exactly half of all 3-digit
numbers have an even number of even digits:

Consider a particular 3-digit number, say 384.  Find the number which 
is the "9's complement" of that number--that is, the number which when 
added to the given number gives a result of 999.  For this example, we 
have 999-384=615.  So 384 and 615 are a "pair" of numbers related by 
the special fact that their sum is 999.

The original number contains two even digits.  We found the other 
number by subtracting the original number from 999.  9 is an odd 
digit; and an odd digit can be expressed as the sum of two digits only 
as the sum of one odd and one even digit.  Therefore, every even digit 
in our original number will correspond to an odd digit in our "new" 
number, while every odd digit in our original number will correspond 
to an even digit in our "new" number:

  3 <==> 6
  8 <==> 1
  4 <==> 5

We can think of pairing up all 900 3-digit numbers with their "9's 
complement" numbers in this manner.  The numbers all contain 3 digits; 
and in each pair, each even digit of one number corresponds to an odd 
digit of the other number and vice versa.  Therefore, in each pair, 
one of the two 3-digit numbers will have an even number of even digits 
and the other will have an odd number of even digits.  And since our 
complete list of pairings contains all 900 3-digit numbers, exactly 
half of those 900 3-digit numbers have an even number of even digits 
and exactly half have an odd number of even digits.

I hope all this helps.  Please write back if you have any further 
questions about any of this.

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

Date: 10/24/2004 at 13:55:45
From: Kevin
Subject: Thank you (3 digit even number problem.)

Thanks, that really helped me understand the problem and a way to the
solution.

Thanks!