In probability, the **trinomial coefficient** (sometimes called the *central trinomial coefficient*) is the number of ways of partitioning a set of objects into three disjoint subsets.

The value of the trinomial coefficient can be calculated as (Hilton et al., 2002; Kuri-Morales & Simari, 2002):

Where:

- ! is a factorial,
- k is the number of objects you want to partition (non-negative integer),
- a, b, c, are the three subsets (also must be non-negative integers),

## Trinomial Coefficient and the Trinomial Theorem

Trinomial coefficients appear in the **trinomial theorem**, which expands a trinomial (x + y + z)^{k} as the algebraic coefficient of x^{a} y^{b} z^{c}:

## Relation to Binomial Coefficient

The trinomial coefficient is a **close relative of the binomial coefficient**:

- The
**trinomial**coefficient appears in the expansion of a trinomial (x + y + z)^{k}and is the number of ways of partitioning three sets. - The
**binomial**coefficient appears in the expansion of a binomial (x + y)^{k}, and is the number of ways of partitioning two sets.

## The Trinomial Triangle

The **trinomial triangle**, an extension of Pascal’s triangle, gives the coefficients of the expansion (1 + x + x^{2})^{k}.

The entries in each row represent “k”. For example, the second row (k = 2) has entries 1 2 3 2 1, giving the expansion

(1 + x + x^{2})^{2} = 1 + 2x + 3x^{2} + 2x^{3} + x^{4}.

## References

Engelward, A. The Analogies between Binomial and Trinomial Coefficients. Retrieved September 29, 2020 from: http://people.math.harvard.edu/~engelwar/MathS305/Trinomial%20Coefficients.pdf

Graham, D. & Allinson, N. (1998). Characterizing Virtual Eigensignatures for General Purpose Face Recognition. Face Recognition From Theory to Application 163, 446-456.

Hilton, P. et al. (2002). Mathematical Vistas: From a Room with Many Windows. Springer.

Kuri-Morales, A. & Simari, G. (2010). Advances in Artificial Intelligence – IBERAMIA 2010. 12th Ibero-American Conference on AI, BahÃa Blanca, Argentina, November 1-5, 2010, Proceedings. Springer Berlin Heidelberg.