How many gifts are there in the song Twelve Days of Christmas?

Day 1: 1 gift

Day 2: 1 + 2 = 3 gifts

Day 3: 1 + 2 + 3 = 6 gifts

…

Day 12: 1 + 2 + 3 + … + 12 = 78 gifts

The number of gifts on day *n* is the *n*th triangular number. The total number of gifts up to and including day *n* is the sum of the first *n* triangular numbers, known as the *n*th tetrahedral number. In the image below, the total number of balls is the fifth tetrahedral number. The number of balls in each layer are triangular numbers. (Image credit: Math is Fun.)

I’ll develop a formula for tetrahedral numbers and continuations of the pattern such as the sum of tetrahedral numbers etc.

First, let T(*n*, 1) = *n*.

Next, let T(*n*, 2) be the *n*th triangular number. So T(*n*, 2) is the sum of the first *n* terms in the sequence T(*i*, 1).

Next, let T(*n*, 3) be the *n*th tetrahedral number. So T(*n*, 3) is the sum of the first *n* terms in the sequence T(*i*, 2).

In general, define T(*n*, *k*) to be the sum of the first *n* terms in the sequence T(*i*, *k*-1). You could think of T(*n*, *k*) as the *n*th *k*-dimensional triangular number. (A tetrahedron is a sort of 3-dimensional triangle. It’s a pyramid whose base is a triangle. T(n,4) would count balls arranged in a sort of 4-dimensional triangle, a simplex in 4 dimensions.)

**Theorem**: T(*n*, *k*) = *n*(*n*+1)(*n*+2) … (*n*+*k*-1)/*k*!

**Corollary**: There are T(12, 3) = 12*13*14/6 = 364 gifts in the Twelve Days of Christmas.

See these notes for a elementary proof by induction.

**Update**: Here’s more advanced proof that uses calculus of finite differences. The more advanced proof requires more background, but it also gives a better idea of how someone might have discovered the formula.

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