∑

Binomial Theorem Expansion Calculator

Full Expansion · Pascal's Triangle · Specific Term Finder

Expand (ax + b)ⁿ completely — every term, binomial coefficients, Pascal's triangle row, and step-by-step breakdown. Supports n up to 20.

Quick Examples

Coefficient a

Variable label

Constant b

Power n (1–20)

Power n

—

exponent

Terms

—

n + 1

Leading Coeff

—

aⁿ

Constant Term

—

bⁿ

Coeff Sum

—

2ⁿ (a=b=1)

Full Expanded Polynomial

Term-by-Term Breakdown

General term: T(k+1) = C(n,k) · (a)ⁿ⁻ᵏ · (b)ᵏ · x^(n-k)

k	C(n,k)	aⁿ⁻ᵏ	bᵏ	Coefficient	Power of x	Term

Pascal's Triangle Row for n = 4

These are the binomial coefficients C(n,0) through C(n,n).

Render Pascal's Triangle up to row n (max 15). The highlighted row shows the binomial coefficients for the selected n.

Show rows 0 through n (max 15)

Pascal's Triangle

Highlighted row = selected n. Each entry = sum of two entries above it.

Row Sums

Row n	Coefficients	Sum = 2ⁿ

What Is the Binomial Theorem?

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)ⁿ into a sum of terms involving powers of a and b. The theorem states:

(a + b)ⁿ = Σ _k=0ⁿ C(n,k) · aⁿ⁻ᵏ · bᵏ

where C(n,k) = n! / (k!(n-k)!) is the binomial coefficient, also written nCk or "n choose k." This elegant formula produces exactly n+1 terms and works for any real numbers a and b.

History: From Pascal to Newton

Special cases of the binomial theorem were known to ancient Indian mathematicians (Pingala, ~200 BC) and later to Omar Khayyam and Chinese mathematician Yang Hui. Blaise Pascal systematized the triangle of coefficients in his Traité du triangle arithmétique (1653), giving us Pascal's Triangle.

Isaac Newton made the landmark generalization in 1665, extending the theorem to fractional and negative exponents. His generalized binomial series (1+x)ⁿ = 1 + nx + n(n-1)/2! · x² + ... converges for |x| < 1 even when n is not a positive integer, underpinning much of classical analysis.

Pascal's Triangle and Binomial Coefficients

Pascal's Triangle is a triangular array where each entry C(n,k) is the sum of the two entries above it: C(n,k) = C(n-1, k-1) + C(n-1, k). Row n of Pascal's Triangle gives exactly the coefficients in the expansion of (a+b)ⁿ. This recurrence makes it easy to compute all coefficients without factorials.

Reference Table: Binomial Coefficients C(n,k)

n	Row (k = 0 to n)	Sum
0	1	1
1	1 1	2
2	1 2 1	4
3	1 3 3 1	8
4	1 4 6 4 1	16
5	1 5 10 10 5 1	32
6	1 6 15 20 15 6 1	64

The General Term Formula

The (k+1)-th term in the expansion of (ax + b)ⁿ is:

T(k+1) = C(n,k) · (ax)ⁿ⁻ᵏ · bᵏ = C(n,k) · aⁿ⁻ᵏ · bᵏ · x^(n-k)

To find the term containing x^r: set n - k = r, giving k = n - r. Then T(n-r+1) = C(n, n-r) · aʳ · bⁿ⁻ʳ · xʳ. To find the middle term (when n is even): k = n/2.

Applications in Probability: The Binomial Distribution

The binomial theorem is the mathematical foundation of the Binomial Distribution in probability. If a random experiment has probability p of success, the probability of exactly k successes in n independent trials is:

P(X = k) = C(n,k) · pᵏ · (1-p)ⁿ⁻ᵏ

Notice this is exactly one term of the binomial expansion of (p + (1-p))ⁿ = 1. The sum of all probabilities over k = 0 to n equals 1, directly verified by the coefficient sum identity C(n,0) + ... + C(n,n) = 2ⁿ when p = 1-p = 1/2.

The Binomial Approximation

For small x, the binomial approximation (1 + x)ⁿ ≈ 1 + nx is extremely useful. This retains only the first two terms of the full expansion. The approximation is accurate when |x| << 1 and is widely applied in:

Physics: Relativistic energy corrections, gravitational potential, optics
Finance: Compound interest approximations for small rates
Engineering: Linearization of nonlinear models near operating points
Combinatorics: Counting paths, subsets, and arrangements in large sets

Combinatorial Identity: Sum of Coefficients = 2ⁿ

Setting a = b = 1 in the binomial theorem gives (1+1)ⁿ = 2ⁿ, so the sum of all binomial coefficients in row n is exactly 2ⁿ. Combinatorially, this equals the total number of subsets of an n-element set — each element is either included or excluded. Setting a = 1, b = -1 gives the alternating sum C(n,0) - C(n,1) + C(n,2) - ... = 0, another important identity.

Frequently Asked Questions

What is the Binomial Theorem?

The Binomial Theorem states that (a+b)^n = Σ_k=0ⁿ C(n,k) · a^(n-k) · b^k, where C(n,k) = n! / (k!(n-k)!) is the binomial coefficient. It produces a polynomial with n+1 terms. This formula applies for any real or complex a, b and any non-negative integer n.

What are binomial coefficients and how do they relate to Pascal's Triangle?

Binomial coefficients C(n,k) count the number of ways to choose k items from n. They form Pascal's Triangle: C(n,k) = C(n-1,k-1) + C(n-1,k). Row n of Pascal's Triangle gives the coefficients C(n,0), C(n,1), ..., C(n,n) used in the expansion of (a+b)^n.

What is the general term formula in the Binomial Theorem?

The (k+1)-th term in the expansion of (a+b)^n is T(k+1) = C(n,k) · a^(n-k) · b^k, for k = 0, 1, ..., n. To find T(3) for example, substitute k=2: T(3) = C(n,2) · a^(n-2) · b^2.

How do you find the term containing x^r in a binomial expansion?

In (ax + b)^n, the general term contains x^(n-k). To find the term with x^r, set n-k = r, so k = n-r. Then T(n-r+1) = C(n, n-r) · a^r · b^(n-r) · x^r. This is valid only when 0 ≤ r ≤ n. Use the "Specific Term" tab to compute this automatically.

What is the sum of all binomial coefficients in the expansion of (a+b)^n?

The sum C(n,0) + C(n,1) + ... + C(n,n) = 2^n. This follows from setting a = b = 1 in the binomial theorem: (1+1)^n = 2^n. Combinatorially, 2^n is the total number of subsets of an n-element set.

Who discovered the Binomial Theorem and when?

Special cases were known to ancient Indian mathematicians (~200 BC). Pascal systematized the coefficient triangle in 1653. Isaac Newton generalized the theorem in 1665 to fractional and negative exponents, creating the generalized binomial series that converges for |x| < 1.

What is the binomial approximation and when is it used?

When |x| << 1, (1+x)^n ≈ 1 + nx. This keeps only the first two terms of the full expansion. It is used in physics for relativistic corrections, in finance for compound interest approximations, and in engineering for linearizing nonlinear systems near an operating point.