Pascal's Pyramid - Ratio Between Coefficients of Same Layer

Ratio Between Coefficients of Same Layer

On each layer of the Tetrahedron, the numbers are simple whole number ratios of the adjacent numbers. This relationship is illustrated for horizontally adjacent pairs on the 4th Layer by the following:

Because the tetrahedron has three-way symmetry, the ratio relation also holds for diagonal pairs (in both directions), as well as for the horizontal pairs shown.

The ratios are controlled by the exponents of the corresponding adjacent terms of the trinomial expansion. For example, one ratio in the illustration above is:

The corresponding terms of the trinomial expansion are:

The following rules apply to the coefficients of all adjacent pairs of terms of the trinomial expansion:

• The exponent of one of the variables remains unchanged (B in this case) and can be ignored.
• For the other two variables, one exponent increases by 1 and one exponent decreases by 1.
  • The exponents of A are 3 and 2 (the larger being in the left term).
  • The exponents of C are 0 and 1 (the larger being in the right term).
• The coefficients and larger exponents are related:
  • 4 × 3 = 12 × 1
  • 4 / 12 = 1 / 3
• These equations yield the ratio: "1:3".

The rules are the same for all horizontal and diagonal pairs. The variables A, B, C will change.

This ratio relationship provides another (somewhat cumbersome) way to calculate tetrahedron coefficients:

The coefficient of the adjacent term equals the coefficient of the current term multiplied by the current-term exponent of the decreasing variable divided by the adjacent-term exponent of the increasing variable.

The ratio of the adjacent coefficients may be a little clearer when expressed symbolically. Each term can have up to six adjacent terms:

where C(x,y,z) is the coefficient and x, y, z are the exponents. In the days before pocket calculators and personal computers, this approach was used as a school-boy short-cut to write out Binomial Expansions without tedious algebraic expansions or clumsy factorial computations.

This relationship will work only if the trinomial expansion is laid out in the non-linear fashion as it is portrayed in the section on the "trinomial expansion connection".