How Many Valentines?

Date: 02/10/2003 at 08:31:34
From: Aman
Subject: Valentines

If everyone in your class gave a Valentine to everyone else in your 
class, how many Valentines would be exchanged? How many Valentines 
would be exchanged if every student in your school exchanged 
Valentines? If all the residents in your city exchanged Valentines? Is 
there a way you could figure it out for any number of people?

Date: 02/10/2003 at 09:53:35
From: Doctor Edwin
Subject: Re: Valentines

Hi, Aman.

Let's start with a small number of students. We'll make a chart and 
see if we can get some idea of what's going on.

If you have just two students, how many Valentines get exchanged? 
A buys one for B, B buys one for A. Each person buys one, so that's 
two.

What if you have three students, A, B, and C? A buys one each for B 
and C, B buys one each for C and A, and C buys one each for A and B. 
Each person buys two, and there are three people, so that's six.

What if there are four people, A, B, C, and D. How many does A buy? 
If there are four people, and they each buy that many, how many get 
bought?

Let's make that chart and see where all this is going:

   # of         How many        Total number
 students     each one buys        bought

    1              0                 0
    2              1                 2
    3              2                 6
    4              3                12

In other words, each person in the room buys as many valentines as 
there are people in the room - ALMOST! You wouldn't buy one for 
yourself, would you? So each person buys as many Valentines as there 
are people in the room MINUS ONE.

So if you have 5 people in the room, then there are 5 * 4 Valentines 
exchanged. If you have 6 people in the room, then there are 6 * 5 
Valentines exchanged.

What if there are 100 people in the room? What if there are N people 
in the room? Can you answer those?

Write back if you need more help!

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