Finding n(n-1)/2 in the Real World

Date: 09/14/2002 at 10:33:49
From: john cook
Subject: Explain a formula

Dear Dr. Math,

How can n(n-1)/2 be explained in common everyday language? When is it 
used?

Thanks,
John

Date: 09/14/2002 at 11:40:59
From: Doctor Ian
Subject: Re: Explain a formula

Hi John,

One place it might come from the real world is this. Suppose you know 
that your company is going to buy some computers, and you need to have 
a direct connection from each one to each other one. (You would 
normally use something like the Internet to hook them up, but direct 
connections are more secure.) 

How many connections will you need? 

If you have one computer, you need zero connections. If you add a 
second computer, you have to add a connection to the first one:

   A ..... B

If you add a third computer, you have to add a connection to each of 
the ones you already had:

   A ..... B
    .     .
     .   .
       C

If you make a table, 

    Computer    Additional connections
    --------    ----------------------
       1            0
       2            1
       3            2
       4            3
       .            .
       .            .
       n           n-1

So if you buy N computers, the number of connections you'll need is

  1 + 2 + 3 + ... + (N-1)

Now, is there a way to figure this out without doing a zillion 
additions?  In fact, there is; see the Dr. Math archives:

   Sum of Numbers 1-500
   http://mathforum.org/library/drmath/view/57990.html 

But we have to make one little adjustment. If you add the first N 
numbers, you get
   
                        N(N+1)
  1 + 2 + 3 + ... + N = ------
                          2

which isn't quite the same formula. But we're adding, not the first N 
numbers, but the first (N-1) numbers. So N becomes (N-1), and (N+1) 
becomes N, which gives us

                            (N-1)((N-1)+1)      
  1 + 2 + 3 + ... + (N-1) = --------------
                                   2

                            (N-1)N
                          = ------
                               2

Note that this kind of 'connections' problem arises in lots of 
different contexts: How many direct flights can you have between 
various cities? How many exchange rates can you have between various 
currencies? How many translators do you need between various native 
speakers?  

Is this what you wanted to know? 

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