Pascal's Pyramid - Trinomial Expansion Connection

Trinomial Expansion Connection

The numbers of the Tetrahedron are derived from trinomial expansion. The nth Layer is the detached coefficient matrix (no variables or exponents) of a trinomial expression (e.g.: A + B + C) raised to the nth power. 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:

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 "!" means factorial (e.g.: n! = 1 × 2 ×...× 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.