## Pascal’s Triangle and the Maths of Christmas

In the Christmas song, “The 12 Days of Christmas”, how many total gifts does my true love give to me? The gifts are:

A partridge in a pear tree,
Two turtle doves,
Three french hens,
Four calling birds,
Five gold rings,
Six geese a-laying
Seven swans a-swimming,
Eight maids a-milking,
Ten lords a-leaping,
Eleven pipers piping,
Twelve drummers drumming.

There are two ways that people usually do this problem:

We can count the total number of each gift.

I get 1 partridge in a pear tree on each of the 12 days. 1 x 12 = 12

2 turtle doves on the last 11 days. 2 x 11 = 22

3 french hens on the last 10 days. 3 x 10 = 30

4 calling birds on the last 9 days. 4 x 9 = 36

5 gold rings on the last 8 days. 5 x 8 = 40

6 geese a-laying on the last 7 days. 6 x 7 = 42

7 swans a-swimming on the last 6 days. 7 x 6 = 42

8 maids a-milking on the last 5 days. 8 x 5 = 40

9 ladies dancing on the last 4 days. 9 x 4 = 36

10 lords a-leaping on the last 3 days. 10 x 3 = 30

11 pipers piping on the last 2 days. 11 x 2 = 22

12 drummers drumming on the last day. 12 x 1 = 12

That is a total of 364 gifts.

Or We could count the number of gifts I get on each day.

On the first day I get 1 gift

On the second day 1 + 2 = 3gifts.

On the third day 1 + 2 + 3 = 6 gifts.

On the fourth day 1 + 2 + 3+ 4 = 10 gifts.

On the fifth day 1 + 2 + 3+ 4 + 5 = 15 gifts.

On the sixth day 1 + 2 + 3+ 4 + 5 + 6 = 21 gifts.

On the seventh day 1 + 2 + 3+ 4 + 5 + 6 + 7 = 28 gifts.

On the eighth day 1 + 2 + 3+ 4 + 5 + 6 + 7 + 8 = 36 gifts.

On the ninth day 1 + 2 + 3+ 4 + 5 + 6 + 7 + 8 + 9 = 45 gifts.

On the tenth day 1 + 2 + 3+ 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55 gifts.

On the eleventh day 1 + 2 + 3+ 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66 gifts.

On the twelfth day 1 + 2 + 3+ 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11+ 12 = 78 gifts.

Adding: 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 = 364 gifts.

We are adding the triangular numbers. The triangular numbers can be found in Pascal’s triangle, so we can use Pascal’s triangle.