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.

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.

The numbers of the tetrahedron are derived from the trinomial expansion. The n^th layer consists of all the coefficients when the trinomial ${\displaystyle A+B+C}$ is raised to the n^th power. The n^th power of the trinomial is expanded by repeatedly multiplying the trinomial by itself:

Each term in the first expression is multiplied by each term in the second expression; and then the coefficients of like terms (same variables and exponents) are added together. Here is the expansion of (A + B + C)⁴:

4A³B¹C⁰ + 12A²B¹C¹ + 12A¹B¹C² + 4A⁰B¹C³ +
6A²B²C⁰ + 12A¹B²C¹ + 6A⁰B²C² +
4A¹B³C⁰ + 4A⁰B³C¹ +

Writing the expansion in this non-linear way shows the expansion in a more understandable way. It also makes the connection with the tetrahedron obvious−the coefficients here match those of layer 4. All the implicit coefficients, variables, and exponents, which are normally not written, are also shown to illustrate another relationship with the tetrahedron. (Usually, "1A" is "A"; "B¹" is "B"; and "C⁰" is "1"; etc.) The exponents of each term sum to the layer number (n), or 4, in this case. More significantly, the value of the coefficients of each term can be computed directly from the exponents. The formula is ⁠(x+y+z)!/x!y!z!⁠ , where x, y, z are the exponents of A, B, C, respectively, and "!" is the factorial, i. e.: ${\displaystyle n!=1\cdot 2\cdot 3\cdots n}$. The exponent formulas for the 4th layer are:

The exponents of each expansion term can be clearly seen and these formulae simplify to the expansion coefficients and the tetrahedron coefficients of layer 4.