Pascal's pyramid

The binomial and trinomial numbers, 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.

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.

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 for the trinomial distribution is: where x, y, z are the number of times each of the three outcomes does occur; n is the number of trials and equals the sum of x+y+z; and PA, PB, PC are the probabilities that each of the three events could occur.

What is the chance that a randomly selected four-person focus group would contain the following voters: 1 for A, 1 for B, 2 for C?

There are 15 different arrangements of four-person focus groups that can be selected.

It is the sample size−a four-person group−and indicates that the coefficients of these arrangements can be found on layer 4 of the tetrahedron.

But the value of these expression is still equal to the coefficients of the 4th layer of the tetrahedron.

And they can be generalized to any layer by changing the sample size (n).

It is "surrounded" by three numbers of the 3rd layer: 6 to the "north", 3 to the "southwest", 3 to the "southeast".

This relationship between adjacent layers comes about through the two-step trinomial expansion process.

Symbolically, the additive relation can be expressed as: where C(x,y,z) is the coefficient of the term with exponents x, y, z and ⁠

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".

The ratios are controlled by the exponents of the corresponding adjacent terms of the trinomial expansion.

The following rules apply to the coefficients of all adjacent pairs of terms of the trinomial expansion: The rules are the same for all horizontal and diagonal pairs.

This ratio relationship provides another (somewhat cumbersome) way to calculate tetrahedron coefficients: The ratio of the adjacent coefficients may be a little clearer when expressed symbolically.

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 the 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".

This relationship is best illustrated by comparing Pascal's triangle down to line 4 with layer 4 of the tetrahedron.

[2] × 1 = 1 1       1 × 4 = 4       4 1       2       1 × 6 = 6      12      6 1       3       3       1 × 4 = 4      12     12      4 1       4       6       4       1 × 1 = The multipliers (1 4 6 4 1) compose line 4 of Pascal's triangle.

This relationship demonstrates the fastest and easiest way to compute the numbers for any layer of the tetrahedron without computing factorials, which quickly become huge numbers.

(Extended precision calculators become very slow beyond tetrahedron layer 200.)

It compares them to the binomial and multinomial expansions and distributions: Arbitrary layer n can be obtained in a single step using the following formula: where b is the radix and d is the number of digits of any of the central multinomial coefficients, that is then wrapping the digits of its result by d(n+1), spacing by d and removing leading zeros.

This method generalised to arbitrary dimension can be used to obtain slices of any Pascal's simplex.

For radix b = 10: In genetics, it is common to use Pascal's pyramid to find out the proportion between different genotypes on the same crossing.

This is done by checking the line that is equivalent to the number of phenotypes (genotypes + 1).