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Binomial Expansion Calculator

Expand (a + b)^n using the binomial theorem and find specific terms.

(a + b)^n Parameters

Find Specific Term

Binomial Theorem

(a + b)^n = Sum[k=0 to n] C(n,k) * a^(n-k) * b^k

Where C(n,k) = n! / (k!(n-k)!)

Binomial Coefficients

C(4,0)1
C(4,1)4
C(4,2)6
C(4,3)4
C(4,4)1

(1 + 1)^4

= 16

Term 3

Coefficient6
a power2
b power2
Value6

All Terms

#C(n,k)a^b^Value
11401.0000
24314.0000
36226.0000
44134.0000
51041.0000

Middle Term(s)

Term 3

6

What Is Binomial Expansion?

Binomial expansion expresses (a+b)n as a sum of terms of the form C(n,k) × an−k × bk for k = 0 to n, where C(n,k) is the binomial coefficient. The calculator takes a, b, and n as inputs and generates the full expanded polynomial with all terms, plus can find any specific term by its position k.

Binomial expansion is fundamental to algebra, combinatorics, and probability. The coefficients follow Pascal's triangle. For (1+x)n with small x, the expansion provides approximations (1+x)n ≈ 1 + nx for |x| ≪ 1 — the basis of binomial approximations in physics and finance.

Binomial Theorem Formula

Binomial Theorem

(a + b)^n = Σ C(n,k) × a^(n−k) × b^k for k=0 to n

Where:

  • a= First term in the binomial
  • b= Second term in the binomial
  • n= Exponent — a non-negative integer
  • C(n,k)= Binomial coefficient: n! / (k!(n−k)!)

How to Use

  1. Enter a and b: The two terms in the binomial.
  2. Enter n: The exponent.
  3. Optionally enter k: To find a specific term position.
  4. Read the expansion: The full expanded form or the specific term is displayed.

Applications

Binomial expansion is used in probability for the binomial distribution mass function. In calculus, series expansions approximate functions near a point. In finance, compound interest formulas (1+r)n expand to show individual period contributions. In physics, relativistic energy expansions use the binomial theorem for low-velocity approximations.

Worked Examples

Expand (x + y)^4

Problem:

Expand (1x + 1y)^4.

Solution Steps:

  1. 1C(4,0)=1: 1x⁴
  2. 2C(4,1)=4: 4x³y
  3. 3C(4,2)=6: 6x²y²
  4. 4C(4,3)=4: 4xy³
  5. 5C(4,4)=1: 1y⁴

Result:

x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

Find Specific Term

Problem:

Find the 3rd term (k=3) of (2 + x)^5.

Solution Steps:

  1. 1C(5,3) = 10
  2. 2a^(5−3) × b³ = 2² × x³ = 4x³
  3. 3Term: 10 × 4x³ = 40x³

Result:

Term k=3: 40x³

Tips & Best Practices

  • ✓The sum of all binomial coefficients in row n equals 2^n — a quick sanity check for your expansion.
  • ✓Pascal's triangle row n gives all C(n,k) coefficients — the calculator computes these using the factorial formula.
  • ✓For large n, the middle terms (k ≈ n/2) have the largest coefficients.
  • ✓The binomial theorem generalizes to multinomials: (a+b+c)^n = Σ (n!/(i!j!k!)) × a^i × b^j × c^k.

Related Calculators

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Pascal's Triangle

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Combination Calculator

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Frequently Asked Questions

C(n,k) = n!/(k!(n−k)!) is the binomial coefficient — the number of ways to choose k items from n. In binomial expansion, it's the coefficient of the kth term and follows Pascal's triangle: each row gives coefficients for (a+b)^n.
With integer n, the expansion has exactly n+1 terms and terminates. For fractional or negative n, the expansion becomes an infinite series (Newton's generalized binomial theorem). This calculator works with non-negative integer n.
Negative values work fine. For (x − y)^n, set b = −y. The signs alternate in the expansion: terms with odd k have negative coefficients.

Sources & References

Last updated: 2026-06-06

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Editorial Note

MyCalcBuddy Editorial Team

This page is maintained as an educational calculator reference.

Source

Formula Source: Handbook of Mathematical Functions

by Abramowitz & Stegun

UpdatedLast reviewed: May 2026
CheckedFormula checks are based on standard references and internal QA review.