Recipe
Extract one term from a multinomial expansion
Use when: you need the coefficient of one specific term in an expanded (x_1+\cdots+x_m)^n, without expanding the whole thing
Goal: the correct multinomial coefficient for the requested term
Taught in The multinomial theorem
Quick recipe
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1
Match each variable's exponent in the target term to its own k_i, in the same order the variables appear in the original sum.
Example
Coefficient of a^2b^2c^2 in (a+b+c)^6: k_1=2 (for a), k_2=2 (for b), k_3=2 (for c).
Why: the multinomial coefficient needs every variable's exponent named individually, not just the total power.
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2
Add every k_i from Step 1 and check the total matches the power n the whole sum is raised to.
Example
2+2+2=6, matching n=6.
- the exponents sum to n exactly — continue to Step 3.
- the exponents don't sum to n — that term doesn't actually appear in the expansion at all; its coefficient is 0, and no formula needs to be applied.
Why: the multinomial theorem only sums over splits of n into exactly these parts; a target term whose exponents don't add up to n was never one of the terms in the first place.
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3
Substitute n and every k_i from Steps 1 and 2 into the multinomial coefficient formula.
Example
\dfrac{6!}{2!\,2!\,2!}.
Why: this single formula generalizes the binomial coefficient to any number of variables, reducing to \binom{n}{k} exactly when m=2.
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4
Evaluate the numerator and denominator factorials, then divide.
Example
\dfrac{720}{8}=90.
Why: this is the actual numeric coefficient the problem asked for.
Common mistakes
- Forgetting one of the variables entirely when there are three or more, especially a variable whose exponent is 0; it still needs a 0!=1 in the denominator.
- Checking that the exponents sum to n only after computing the formula, instead of before, wasting the computation on a term that was never actually present.
- Mixing up which exponent belongs to which variable when the term isn't written in the same order as the original sum.
- Treating 0! as 0 instead of 1, which silently zeroes out an otherwise-correct computation.