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Trinomial Coefficient & Theorem: Definition

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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):
trinomial coefficient 2


  • ! 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 xa yb zc:

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 + x2)k.
trinomial triangle
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 + x2)2 = 1 + 2x + 3x2 + 2x3 + x4.


Engelward, A. The Analogies between Binomial and Trinomial Coefficients. Retrieved September 29, 2020 from:
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, Bahía Blanca, Argentina, November 1-5, 2010, Proceedings. Springer Berlin Heidelberg.

Stephanie Glen. "Trinomial Coefficient & Theorem: Definition" From Elementary Statistics for the rest of us!

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