Twelve Days of Christmas Gifts

Date: 12/06/2001 at 18:26:25
From: Katie
Subject: 7th grade math

How many presents did the person receive in the song "The 12 Days of 
Christmas"?

Date: 12/07/2001 at 15:58:16
From: Doctor Ian
Subject: Re: 7th grade math

Hi Katie,

On the first day, there is one gift:

  1

On the second day, there is another one of that gift, and two more:

  1
  1 2

On the third day,

  1 
  1 2
  1 2 3

and so on, up to 12 days:

  1
  1 2
  1 2 3
  1 2 3 4
  1 2 3 4 5
  1 2 3 4 5 6 
  1 2 3 4 5 6 7
  1 2 3 4 5 6 7 8
  1 2 3 4 5 6 7 8 9
  1 2 3 4 5 6 7 8 9 10
  1 2 3 4 5 6 7 8 9 10 11
  1 2 3 4 5 6 7 8 9 10 11 12
  ^ ^ ^                   
  | | |
  | | +---------------------------- 10 * 3
  | +------------------------------ 11 * 2
  +-------------------------------- 12 * 1


So that's

  12*1 + 11*2 + 10*3 + 9*4 + ... 4*9 + 3*10 + 2*11 + 1*12
 
Note that some of the terms are the same:

  12*1 + 11*2 + 10*3 + 9*4 + ... 4*9 + 3*10 + 2*11 + 1*12
    \      \      \     \_________/     /      /      /
     \      \      \___________________/      /      /
      \      \_______________________________/      /
       \___________________________________________/                         


That's convenient, because it means you only have to add up half the 
terms:

  2 * [12*1 + 11*2 + 10*3 + 9*4 * 8*5 * 7*6]

Can you take it from here?

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