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Sum of Integers in a Set


Date: 06/19/2001 at 01:32:00
From: C. Martin
Subject: The sum of integers in this set

Consider the set of all four-digit integers, each of which is formed 
using the digits 1,2,3,5, or 7 at most once. Find the sum of the 
integers in this set.

I realize there are 120 possible combinations of four-digit numbers.  
I can write out all 120 and then add them together, but that just 
seems too long. Is there an easier (more efficient) way to solve this 
problem?  

Thanks for your help.


Date: 06/19/2001 at 11:59:31
From: Doctor Ian
Subject: Re: The sum of integers in this set

Hi C, 

Sometimes the way to attack a large problem is to look at a smaller 
problem first. Suppose I want to find the sum of all the three-digit 
numbers that I can make from digits a, b, and c.  I can write down the 
numbers:

  abc
  acb
  bac
  bca
  cab
  cba

I can expand them this way:

  a*10^2 + b*10^1 + c*10^0
  a*10^2 + c*10^1 + b*10^0
  b*10^2 + a*10^1 + c*10^0
  b*10^2 + c*10^1 + a*10^0
  c*10^2 + a*10^1 + b*10^0
  c*10^2 + b*10^1 + a*10^0

Now if I add these up, I get

  2(a+b+c)*10^2 + 2(a+b+c)*10^1 + 2(a+b+c)*10^0 

which simplifies to 

  2(a+b+c)(10^2 + 10^1 + 10^0)

Does this shed some light on your problem? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Permutations and Combinations
High School Sets

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