Figure out how many gifts are given in the 12 Days of Christmas with Pascal’s Triangle!
Have you ever stopped to think about the number of gifts that are actually given in the traditional Christmas carol “The 12 Days of Christmas”? Now that you are thinking about it, there’s an easy way to figure it out using Pascal’s Triangle.
Learn about Pascal’s Triangle and make a Christmas tetrahedron with this 12 Days of Christmas challenge!
What is Pascal’s Triangle?
Pascal’s Triangle is named after Blaise Pascal, a famous French Mathematician and Philosopher. However, the triangle and concept date back centuries in India, Iran, China, Germany, and Italy.
Pascal’s Triangle is a triangular array of binomial coefficients determined by binomial expansion. What is all of this crazy math talk?! Well, binomials are used in algebra and look like 4x+10 or 5x+2.
Basically, Pascal’s Triangle shows you the probability of any combination. When you look at the triangle, you’ll see the expansion of powers of a binomial where each number in the triangle is the sum of the two numbers above it.
Patterns, Patterns, Patterns!
There are so many neat patterns in Pascal’s Triangle. The first diagonal of the triangle just contains “1”s while the next diagonal has numbers in numerical order. The third diagonal has triangular numbers and the fourth has tetrahedral numbers.
If you add up each row of Pascal’s Triangle, you’ll find the powers of 2 since they double each time. If you squish all of the numbers in each row together, you’ll find exponents of 11.
You’ll also enjoy: Pascal’s Triangle Christmas Tree Patterns Math Activity
Pascal’s Triangle is also symmetrical! If you were to fold the triangle in half, the numbers on the right side are identical to the numbers on the left side.
The pattern that I think is super cool is the Sierpinski Triangle, which can be found if you color all of the odd numbers in Pascal’s Triangle.
12 Days of Christmas
Take a peek at the lyrics of the 12 Days of Christmas:
Traditional with additions by Frederic Austin
On the first day of Christmas, my true love sent to me
A partridge in a pear tree.
On the second day of Christmas, my true love sent to me
Two turtle doves and a partridge in a pear tree.
On the third day of Christmas, my true love sent to me
Three French hens, two turtle doves, and a partridge in a pear tree.
On the fourth day of Christmas, my true love sent to me
Four calling birds, three French hens, two turtle doves, and a partridge in a pear tree.
On the fifth day of Christmas, my true love sent to me
Five gold rings, four calling birds, three French hens, two turtle doves, and a partridge in a pear tree.
On the sixth day of Christmas, my true love sent to me
Six geese a-laying, five gold rings, four calling birds, three French hens, two turtle doves, and a partridge in a pear tree.
On the seventh day of Christmas, my true love sent to me
Seven swans a-swimming, six geese a-laying, five gold rings, four calling birds, three French hens, two turtle doves, and a partridge in a pear tree.
On the eighth day of Christmas, my true love sent to me
Eight maids a-milking, seven swans a-swimming, six geese a-laying, five gold rings, four calling birds, three French hens, two turtle doves, and a partridge in a pear tree.
On the ninth day of Christmas, my true love sent to me
Nine ladies dancing, eight maids a-milking, seven swans a-swimming, six geese a-laying, five gold rings, four calling birds, three French hens, two turtle doves, and a partridge in a pear tree.
On the tenth day of Christmas, my true love sent to me
Ten lords a-leaping, nine ladies dancing, eight maids a-milking, seven swans a-swimming, six geese a-laying, five gold rings, four calling birds, three French hens, two turtle doves, and a partridge in a pear tree.
On the eleventh day of Christmas, my true love sent to me
Eleven pipers piping, ten lords a-leaping, nine ladies dancing, eight maids a-milking, seven swans a-swimming, six geese a-laying, five gold rings, four calling birds, three French hens, two turtle doves, and a partridge in a pear tree.
On the twelfth day of Christmas, my true love sent to me
Twelve drummers drumming, eleven pipers piping, ten lords a-leaping, nine ladies dancing, eight maids a-milking, seven swans a-swimming, six geese a-laying, five gold rings, four calling birds, three French hens, two turtle doves, and a partridge in a pear tree.
How many gifts were sent in the 12 Days of Christmas?
Let’s break it down using Pascal’s triangle.
On the First Day of Christmas – A Partridge in a Pear Tree
On the first (purple triangle) day of Christmas, 1 partridge in a pear tree was given (red triangle) for a combined total of 1 gift given overall (blue triangle).
On the Second Day of Christmas – Two Turtle Doves
On the second (purple triangle) day of Christmas, 1 partridge in a pear tree and 2 turtle doves are given for a total of 3 gifts given that day (red triangle). The combined total of gifts given is 4 (blue triangle).
On the Third Day of Christmas – Three French Hens
On the third (purple triangle) day of Christmas, 1 partridge in a pear tree, 2 turtle doves, and 3 French hens are given for a total of 6 gifts given that day (red triangle). The combined total of gifts given is 10 (blue triangle).
On the Fourth Day of Christmas – Four Calling Birds
On the fourth (purple triangle) day of Christmas, 1 partridge in a pear tree, 2 turtle doves, 3 French hens, and 4 calling birds are given for a total of 10 gifts given that day (red triangle). The combined total of gifts given is 20 (blue triangle).
On the Fifth Day of Christmas – Five Golden Rings
On the fifth (purple triangle) day of Christmas, 1 partridge in a pear tree, 2 turtle doves, 3 French hens, 4 calling birds, and 5 golden rings are given for a total of 15 gifts given that day (red triangle). The combined total of gifts given is 35 (blue triangle).
On the Sixth Day of Christmas – Six Geese Laying
On the sixth (purple triangle) day of Christmas, 1 partridge in a pear tree, 2 turtle doves, 3 French hens, 4 calling birds, 5 golden rings, and 6 geese a-laying are given for a total of 21 gifts given that day (red triangle). The combined total of gifts given is 56 (blue triangle).
On the Seventh Day of Christmas – Seven Swans Swimming
On the seventh (purple triangle) day of Christmas, 1 partridge in a pear tree, 2 turtle doves, 3 French hens, 4 calling birds, 5 golden rings, 6 geese a-laying, and 7 swans a-swimming are given for a total of 28 gifts given that day (red triangle). The combined total of gifts given is 84 (blue triangle).
On the Eighth Day of Christmas – Eight Maids Milking
On the eighth (purple triangle) day of Christmas, 1 partridge in a pear tree, 2 turtle doves, 3 French hens, 4 calling birds, 5 golden rings, 6 geese a-laying, 7 swans a-swimming, and 8 maids a-milking are given for a total of 36 gifts given that day (red triangle). The combined total of gifts given is 120 (blue triangle).
On the Ninth Day of Christmas – Nine Ladies Dancing
On the ninth (purple triangle) day of Christmas, 1 partridge in a pear tree, 2 turtle doves, 3 French hens, 4 calling birds, 5 golden rings, 6 geese a-laying, 7 swans a-swimming, 8 maids-a-milking, and 9 ladies dancing are given for a total of 45 gifts given that day (red triangle). The combined total of gifts given is 165 (blue triangle).
On the Tenth Day of Christmas – Ten Lords Leaping
On the tenth (purple triangle) day of Christmas, 1 partridge in a pear tree, 2 turtle doves, 3 French hens, 4 calling birds, 5 golden rings, 6 geese a-laying, 7 swans a-swimming, 8 maids-a-milking, 9 ladies dancing, and 10 lords a-leaping are given for a total of 55 gifts given that day (red triangle). The combined total of gifts given is 220 (blue triangle).
On the Eleventh Day of Christmas – Eleven Pipers Piping
On the eleventh (purple triangle) day of Christmas, 1 partridge in a pear tree, 2 turtle doves, 3 French hens, 4 calling birds, 5 golden rings, 6 geese a-laying, 7 swans a-swimming, 8 maids-a-milking, 9 ladies dancing, 10 lords a-leaping, and 11 pipers piping are given for a total of 66 gifts given that day (red triangle). The combined total of gifts given is 286 (blue triangle).
On the Twelfth Day of Christmas – Twelve Drummers Drumming
On the twelfth (purple triangle) day of Christmas, 1 partridge in a pear tree, 2 turtle doves, 3 French hens, 4 calling birds, 5 golden rings, 6 geese a-laying, 7 swans a-swimming, 8 maids-a-milking, 9 ladies dancing, 10 lords a-leaping, 11 pipers piping, and 12 drummers drumming are given for a total of 78 gifts given that day (red triangle). The combined total of gifts given is 364 (blue triangle).
Wowza! That’s a lot of gifts!
The Diagonals of Pascal’s Triangle
The second diagonal in Pascal’s triangle (purple triangles) is in numerical order and relates to the 12 Days of Christmas carol by showing us both the day and the amount of the new gift given that day.
The third diagonal is made of triangular numbers (red triangles) and shows us the total number of gifts given on each day.
The fourth diagonal (blue triangles) is made of tetrahedral numbers and shows the combined total of gifts given so far during the 12 Days of Christmas.
A tetrahedral number represents a pyramid with a triangular base and three sides. Just imagine a giant pyramid made of
- 12 partridges in pear trees (12 days x 1 given each day)
- 22 turtle doves (11 days x 2 given each day)
- 30 French hens (10 days x 3 given each day)
- 36 calling birds (9 days x 4 given each day)
- 40 golden rings (8 days x 5 given each day)
- 42 geese a-laying (7 days x 6 given each day)
- 42 swans a-swimming (6 days x 7 given each day)
- 40 maids milking (5 days x 8 given each day)
- 36 ladies dancing (4 days x 9 given each day)
- 30 lords a-leaping (3 days x 10 given each day)
- 22 pipers piping (2 days x 11 given each day)
- 12 drummers drumming (1-day x 12 given that day)
For a grand total of 364 total gifts. It’s probably a good thing there are only 12 drummers with the amount of noise they probably make!
Fill in your own Pascal’s Triangle Christmas Tree and download some free worksheets!
12 Days of Christmas Pascal’s Triangle Challenge
Challenge: Can you use the triangular numbers diagonal to build a “12 days of Christmas” tetrahedron with 364 pom-pom balls?
We used 1/2-inch pom-pom balls and followed the third diagonal of triangular numbers to build our “12 Days of Christmas” tetrahedron.
It was a pretty challenging feat since the pom pom balls wanted to roll, but if you go slowly and make sure to count the right amount per level, you’ll end with a 364 pom-pom ball tetrahedron that represents the total amount of gifts given during the 12 days of Christmas carol!
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Christmas STEAM
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Meet Toni, the Maker Mom behind Our Family Code
Hey there, I’m Toni! I’m a software engineer and Maker Mom that finds my joy in unleashing my children’s curiosity by exploring STEAM concepts with my fantastic five!
When I’m not chasing toddlers or raising tweens, you can find me tearing things up and putting them back together over here at Our Family Code.
I am the owner and content creator of multiple educational websites designed to increase access to STEAM & STEM education with a focus on teaching computer science and coding to kids of all ages!
You can also find out more about me by visiting ToniGardner.com!
