The _{14}C_{3} gifts of Christmas?

This occurred to me a while ago, but after Christmas, and I decided that it couldn't wait until December. (And I'd likely forget about it.)

While coming up with mathematical formulas and computer code for calculating this number, I overlooked a a very reliable reference tool: **Pascal's Triangle**.

It has more uses than simply expanding polynomials because of its many properties.

Some of these properties are as follows:

*sum of the consecutive triangle numbers*.

**Combinations**designated by the notation

_{n}C

_{r}where

*n*is the row and

*r*is the element of that row.

What this means is that for any given day in that song *(The Twelve Days of Christmas)*:

_{n}C

_{1}refers to the day we're up to. On Day 7,

_{7}C

_{1}is 7.

_{n+1}C

_{2}refers to the total number of gifts given on the

*nth*day. On Day 7,

_{8}C

_{2}is 28.

_{n+2}C

_{3}refers to the total number of gifts given altogether up to the

*nth*day. On Day 7,

_{9}C

_{3}is 84.

Applying this to the 12th day of the song:

_{12}C

_{1}is 12.

_{13}C

_{2}is 78.

_{14}C

_{3}is 364.

And I could've been finished a whole lot sooner. But I wouldn't have gotten a recursive comic out of that.

One last thing: Did you ever try to make a poster of **Pascal's Triangle**? Have your students tried to do it for a math fair? The numbers start to get really big in the middle, really fast. (That's a post for another day.) But that's also why the poster in this comic is so large! Otherwise, it wouldn't be readable (and I had to modify the "364" so you could find it easily!).

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