Pascal's pyramid

In mathematics, Pascal's pyramid is a three-dimensional arrangement of the coefficients of the trinomial expansion and the trinomial distribution. Pascal's pyramid is the three-dimensional analog of the two-dimensional Pascal's triangle, which contains the binomial coefficients that appear in the binomial expansion and the binomial distribution. The binomial and trinomial coefficients, expansions, and distributions are subsets of the multinomial constructs with the same names.

Structure of the tetrahedron

Because the tetrahedron is a three-dimensional object, displaying it on a piece of paper, a computer screen, or other two-dimensional medium is difficult. Assume the tetrahedron is divided into a number of levels, floors, slices, or layers. The top layer (the apex) is labeled "Layer 0". Other layers can be thought of as overhead views of the tetrahedron with the previous layers removed. The first six layers are as follows:

The layers of the tetrahedron have been deliberately displayed with the point down so that they are not individually confused with Pascal's triangle.

Overview of the tetrahedron

There is three-way symmetry of the numbers in each layer.

The number of terms in the nth layer is the (n + 1)th triangular number: ⁠

(

n

+

1

)

(

n

+

2

)

2

{\displaystyle {\frac {(n+1)(n+2)}{2}}}

⁠.