Parallels To Pascal's Triangle and Multinomial Coefficients
This table summarizes the properties of the trinomial expansion and the trinomial distribution, and it compares them to the binomial and multinomial expansions and distributions:
| Type of polynomial | bi-nomial | tri-nomial | multi-nomial |
| Order of polynomial | 2 | 3 | m |
| Example of polynomial | A+B | A+B+C | A+B+C+...+M |
| Geometric structure (1) | triangle | tetrahedron | m-simplex |
| Element structure | line | layer | group |
| Symmetry of element | 2-way | 3-way | m-way |
| Number of terms per element | n+1 | (n+1) × (n+2) / 2 | (n+1) × (n+2) ×...× (n+m−1) / (m−1) |
| Sum of values per element | 2n | 3n | mn |
| Example of term | AxBy | AxByCz | AxByCz...Mm |
| Sum of exponents, all terms | n | n | n |
| Coefficient equation (2) | n! / (x! × y!) | n! / (x! × y! × z!) | n! / (x1! × x2! × x3! ×...× xm!) |
| Sum of coefficients "above" | 2 | 3 | m |
| Ratio of adjacent coefficients | 2 | 6 | m × (m−1) |
(1) A simplex is the simplest linear geometric form that exists in any dimension. Tetrahedrons and triangles are examples in 3 and 2 dimensions, respectively.
(2) The formula for the binomial coefficient is usually expressed as: n! / (x! × (n−x)!); where n−x = y.
Read more about this topic: Pascal's Pyramid
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