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.
- ! 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
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.
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: http://people.math.harvard.edu/~engelwar/MathS305/Trinomial%20Coefficients.pdf
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Stephanie Glen. "Trinomial Coefficient & Theorem: Definition" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/trinomial-coefficient-theorem-definition/
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