Recipe

Extract one term from a binomial expansion

Use when: you need the coefficient of one specific term in an expanded $(a+b)^n$ , without expanding the whole thing

Goal: the correct numeric coefficient of the requested term, including any constant riding along with a scaled variable

Taught in The binomial theorem generalized

Quick recipe

Match the exponents to $n$ and $k$
Compute the binomial coefficient $\binom{n}{k}$
Check whether either variable carries its own leading coefficient
Multiply through for the final coefficient

1
Read the two exponents in the target term; $n$ is the power the whole binomial is raised to, and $k$ is the exponent on the second variable.

Example

Coefficient of $x^4y^2$ in $(x+y)^6$ : $n=6$ , and $k=2$ (the exponent on $y$ ).

Why: $n$ and $k$ are only correct once every exponent has been matched to the theorem's own labeling, not guessed from the term's appearance.
2
Apply $\binom{n}{k}$ with the values found in Step 1.

Example

$\binom{6}{2}=15$ .

Why: this is the plain counting part of every binomial term, the number of ways to choose which factors contribute the second variable.
3
Look at the original binomial expression, not just the target term.

Example

For $(3x+y)^5$ , the coefficient of $x^3y^2$ : $x$ carries the coefficient $3$ , and $x$ 's own exponent in this term is $3$ , so the extra factor is $3^3=27$ .
- the binomial is plain, like $(a+b)^n$ , with no leading number on either variable — the count from Step 2 is already the final coefficient; skip to Step 4.
- one variable carries its own coefficient, like the $3$ in $(3x+y)^n$ — that coefficient must be raised to its own exponent and multiplied in during Step 4.
Why: the binomial coefficient alone only counts arrangements of factors; a genuine leading coefficient contributes its own multiplicative factor for every copy the exponent calls for.
4
Multiply the binomial coefficient from Step 2 by any extra factor identified in Step 3.

Example

$\binom{5}{2}\cdot 3^3 = 10\cdot 27=270$ .

Why: the binomial coefficient by itself is only the counting part; the requested coefficient also includes any constants the terms themselves carry.

Common mistakes

Reporting the plain binomial coefficient $\binom{n}{k}$ as the final answer when one of the variables actually had a leading coefficient to account for.
Raising a leading coefficient to the wrong exponent, using $k$ instead of the exponent that variable itself carries in the target term.
Swapping which variable's exponent is $k$ , especially when the target term isn't written in the same variable order as the original binomial.
Forgetting that $n$ is the power on the whole binomial, not the exponent on either individual variable.