Pascals Triangle

(Click on the comic if you can't see the full image.)
(C)Copyright 2015, C. Burke.

The ₁₄C₃ 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:

The "zeroth" element of each row is the number 1.

The first element of each row is the number of the row (keeping in mind that the top row is Row 0).

The second element of each row is are the consecutive triangle numbers, the sum of the consecutive numbers before it.

Which makes the third element of each row the sum of the consecutive triangle numbers.

And, finally, the position of each element in Pascal's Triangle corresponds to the number of Combinations designated by the notation _nC_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):

_nC₁ refers to the day we're up to. On Day 7, ₇C₁ is 7.

_n+1C₂ refers to the total number of gifts given on the nth day. On Day 7, ₈C₂ is 28.

_n+2C₃ refers to the total number of gifts given altogether up to the nth day. On Day 7, ₉C₃ is 84.

Applying this to the 12th day of the song:

₁₂C₁ is 12.

₁₃C₂ is 78.

₁₄C₃ 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!).