Binomial Theorem Formula

The binomial theorem gives the expansion of (a + b)^n as a sum of terms involving binomial coefficients: (a+b)^n = sum of C(n,k) * a^(n-k) * b^k.

The Formula

(a+b)n=∑k=0n(nk)an−kbk(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

When to use: Each term of (a+b)n(a+b)^n picks 'aa' or 'bb' from each factor. (nk)\binom{n}{k} counts how many ways to pick kk bb's.

Quick Example

(a+b)3=a3+3a2b+3ab2+b3(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 Coefficients 1,3,3,1 are row 3 of Pascal's triangle.

Notation

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!} is the binomial coefficient ('nn choose kk'). ∑\sum denotes summation from k=0k = 0 to nn.

What This Formula Means

The binomial theorem gives the expansion of (a + b)^n as a sum of terms involving binomial coefficients: (a+b)^n = sum of C(n,k) * a^(n-k) * b^k. Each coefficient (nk)\binom{n}{k} counts the number of ways to choose kk copies of bb from nn factors.

Each term of (a+b)n(a+b)^n picks 'aa' or 'bb' from each factor. (nk)\binom{n}{k} counts how many ways to pick kk bb's.

Formal View

∀a,b∈R,  ∀n∈N:(a+b)n=∑k=0n(nk)an−kbk\forall a, b \in \mathbb{R},\; \forall n \in \mathbb{N}: (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k, where (nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}.

Worked Examples

Example 1

medium
Expand (x+2)3(x + 2)^3 using the Binomial Theorem.

Answer

x3+6x2+12x+8x^3 + 6x^2 + 12x + 8

First step

1
Step 1: Apply (a+b)3=a3+3a2b+3ab2+b3(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 with a=xa = x, b=2b = 2.

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Example 2

hard
Find the coefficient of x3x^3 in the expansion of (2x+3)5(2x + 3)^5.

Example 3

medium
Find the coefficient of x4x^4 in (1+x)7(1+x)^7.

Common Mistakes

  • Distributing the exponent as (a+b)n=an+bn(a+b)^n=a^n+b^n - every cross term with (nk)\binom{n}{k} is required.
  • Letting the aa and bb exponents not sum to nn - in each term an−kbka^{n-k}b^k the powers must total nn.
  • Forgetting the coefficient on a chosen term - the term is (nk)an−kbk\binom{n}{k}a^{n-k}b^k, not just an−kbka^{n-k}b^k.

Why This Formula Matters

It converts a brutal repeated multiplication into a one-line, term-by-term formula and links algebra to counting (Pascal's triangle, combinations). It is also the only fast way to extract one specific term of a high power. Recognizing it by "Am I raising a two-term expression to a whole-number power and want its expansion or a single term?" — rather than by familiar numbers — is what lets a student tell it apart from foil / polynomial multiplication and binomial coefficient alone and perfect-square identity in a mixed problem set.

Frequently Asked Questions

What is the Binomial Theorem formula?

The binomial theorem gives the expansion of (a + b)^n as a sum of terms involving binomial coefficients: (a+b)^n = sum of C(n,k) * a^(n-k) * b^k. Each coefficient (nk)\binom{n}{k} counts the number of ways to choose kk copies of bb from nn factors.

How do you use the Binomial Theorem formula?

Each term of (a+b)n(a+b)^n picks 'aa' or 'bb' from each factor. (nk)\binom{n}{k} counts how many ways to pick kk bb's.

What do the symbols mean in the Binomial Theorem formula?

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!} is the binomial coefficient ('nn choose kk'). ∑\sum denotes summation from k=0k = 0 to nn.

Why is the Binomial Theorem formula important in Math?

It converts a brutal repeated multiplication into a one-line, term-by-term formula and links algebra to counting (Pascal's triangle, combinations). It is also the only fast way to extract one specific term of a high power. Recognizing it by "Am I raising a two-term expression to a whole-number power and want its expansion or a single term?" — rather than by familiar numbers — is what lets a student tell it apart from foil / polynomial multiplication and binomial coefficient alone and perfect-square identity in a mixed problem set.

What do students get wrong about Binomial Theorem?

The procedure for binomial theorem is the easy part; the trap is distributing the exponent as (a+b)n=an+bn(a+b)^n=a^n+b^n. Asking "Am I raising a two-term expression to a whole-number power and want its expansion or a single term?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Binomial Theorem formula?

Before studying the Binomial Theorem formula, you should understand: binomial coefficient, exponents.

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