Associated Topics || Dr. Math Home || Search Dr. Math

### 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?

```

```
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

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search