Trinomial Coefficient — Definition, Formula & Examples

A trinomial coefficient is a number that appears when you expand a power of a three-term expression (trinomial), generalizing the familiar binomial coefficient to three variables instead of two.

The trinomial coefficient $\binom{n}{k_1, k_2, k_3}$ is the multinomial coefficient $\frac{n!}{k_1!\, k_2!\, k_3!}$ where $k_1 + k_2 + k_3 = n$ and $k_1, k_2, k_3 \geq 0$ . It counts the number of ways to partition $n$ objects into three groups of sizes $k_1$ , $k_2$ , and $k_3$ .

Key Formula

\binom{n}{k_1, k_2, k_3} = \frac{n!}{k_1!\, k_2!\, k_3!}

Where:

$n$ = The exponent of the trinomial
$k_1, k_2, k_3$ = Non-negative integers that sum to n, representing the exponents of the three terms

How It Works

When you expand

(a + b + c)^n

, each term has the form

\binom{n}{k_1, k_2, k_3} a^{k_1} b^{k_2} c^{k_3}

, where the exponents sum to

n

. To find the coefficient of a particular term, identify the three exponents and compute

\frac{n!}{k_1!\, k_2!\, k_3!}

. This works exactly like the binomial theorem but with an extra variable. The total number of terms in the expansion equals

\binom{n+2}{2}

Worked Example

Problem: Find the coefficient of

a^2 b c

in the expansion of

(a + b + c)^4

Identify exponents: The term

a^2 b c

has exponents

k_1 = 2

k_2 = 1

k_3 = 1

. Check:

2 + 1 + 1 = 4 = n

k_1 = 2,\; k_2 = 1,\; k_3 = 1

Apply the formula: Substitute into the trinomial coefficient formula.

\frac{4!}{2!\, 1!\, 1!} = \frac{24}{2 \cdot 1 \cdot 1} = 12

Answer: The coefficient of

a^2 b c

(a + b + c)^4

12

Why It Matters

Trinomial coefficients appear in probability when outcomes split three ways, such as draws with three categories. They also arise in combinatorics courses and in expanding expressions with three or more terms, which is common in multivariable algebra.

Common Mistakes

Mistake: Using the binomial coefficient formula

\frac{n!}{k!(n-k)!}

with only one of the exponents, ignoring the third.

Correction: You must divide

n!

by the factorials of all three exponents:

\frac{n!}{k_1!\, k_2!\, k_3!}

. The binomial coefficient is the special case where

k_3 = 0