Multinomial Theorem Calculator

Find the coefficient of any term in (a₁x₁ + a₂x₂ + … + aₘxₘ)ⁿ with the multinomial theorem, plus the number of terms and the sum of the coefficients.

Terms inside the bracket	2x + 3y − z
Power n	6
Powers of each letter in the term you want	2, 3, 1

Terms inside the bracket

Each term is a number times one letter, like 2x + 3y − z or a + b + c + d.

Power n

Powers of each letter in the term you want

In the same order as the letters; they must add to n.

Coefficient of x²y³z

−6480

In (2x + 3y − z)⁶ the term in x²y³z is −6480x²y³z. The expansion has 28 terms, and its coefficients add to (4)⁶ = 4096.

Exact. Checked exactly.

Method: Multinomial theorem
Worked out with: AllCalcHub exact integers
Check: Every step recomputed with exact fractions

Multinomial coefficient: 60
Number of terms C(n + m − 1, m − 1): 28
Sum of all coefficients: 4096

About the Multinomial Theorem Calculator

The multinomial theorem extends the binomial theorem to more than two terms: the coefficient of x₁^k₁ ⋯ xₘ^kₘ in (x₁ + ⋯ + xₘ)ⁿ is n!/(k₁! ⋯ kₘ!), where the k's add to n. With numbers in front, multiply by a₁^k₁ ⋯ aₘ^kₘ.

The expansion has C(n + m − 1, m − 1) terms, and putting every letter equal to 1 shows the coefficients add to (a₁ + ⋯ + aₘ)ⁿ.

The formula

coefficient = n!/(k₁! k₂! ⋯ kₘ!) · a₁^k₁ ⋯ aₘ^kₘ

Worked example

Coefficient of x²y³z in (2x + 3y − z)⁶

6!/(2! 3! 1!) = 60
60 × 2² × 3³ × (−1) = −6480

Questions people ask

How is it related to the binomial theorem?

With two terms, n!/(k!(n − k)!) is the binomial coefficient C(n, k).

Why C(n + m − 1, m − 1) terms?

It counts the ways to split n into m whole-number powers (stars and bars).