# Binomial Expansion Calculator

Expand (a + b)^n using the binomial theorem with Pascal's triangle coefficients. Shows all terms for exponents up to 12.

## What this calculates

Expand any expression of the form (a + b)^n using the binomial theorem. Enter values for a, b, and the exponent n (up to 12), and see every term with its binomial coefficient. For example, (2 + 3)^4 expands to 16 + 96 + 216 + 216 + 81 = 625.

## Inputs

- **Value of a** — The first term in (a + b)^n.
- **Value of b** — The second term in (a + b)^n.
- **Exponent (n)** — min 0, max 12 — The power to raise (a + b) to. Maximum 12.

## Outputs

- **Full Expansion** — formatted as text — Each term of the binomial expansion.
- **Binomial Coefficients** — formatted as text — The C(n, k) values from Pascal's triangle.
- **Numeric Result** — The computed value of (a + b)^n.
- **Number of Terms** — How many terms in the expansion (always n + 1).

## Details

**The Binomial Theorem**

The binomial theorem states that:

**(a + b)^n = Σ C(n, k) x a^(n-k) x b^k** for k = 0 to n

where C(n, k) = n! / (k! x (n-k)!) is the binomial coefficient, also written as "n choose k."

**Pascal's Triangle**

The binomial coefficients form Pascal's triangle. Each row gives the coefficients for a particular power:
- n=0: 1
- n=1: 1, 1
- n=2: 1, 2, 1
- n=3: 1, 3, 3, 1
- n=4: 1, 4, 6, 4, 1

Each number is the sum of the two numbers directly above it.

**Worked Example: (x + 2)^3**

Using C(3,0)=1, C(3,1)=3, C(3,2)=3, C(3,3)=1:

(x + 2)^3 = 1 x x^3 + 3 x x^2 x 2 + 3 x x x 4 + 1 x 8 = x^3 + 6x^2 + 12x + 8

**Key Properties**

- The expansion always has n + 1 terms
- Coefficients are symmetric: C(n, k) = C(n, n-k)
- The sum of all coefficients equals 2^n (set a = b = 1)

## Frequently Asked Questions

**Q: What is the binomial theorem?**

A: The binomial theorem gives a formula for expanding (a + b)^n into a sum of terms. Each term has the form C(n, k) x a^(n-k) x b^k, where C(n, k) is the binomial coefficient 'n choose k.' It works for any non-negative integer exponent n.

**Q: How many terms does a binomial expansion have?**

A: A binomial expansion of (a + b)^n always has exactly n + 1 terms. For example, (a + b)^5 has 6 terms, corresponding to k = 0, 1, 2, 3, 4, and 5.

**Q: What is Pascal's triangle?**

A: Pascal's triangle is a triangular array where each entry is the sum of the two entries above it. The nth row (starting from row 0) gives the binomial coefficients C(n, 0), C(n, 1), ..., C(n, n). Row 4, for instance, is 1, 4, 6, 4, 1.

**Q: Does the binomial theorem work for negative or fractional exponents?**

A: The standard binomial theorem works for non-negative integers. For negative or fractional exponents, the generalized binomial theorem produces an infinite series rather than a finite polynomial. This calculator handles non-negative integer exponents up to 12.

---

Source: https://vastcalc.com/calculators/math/binomial-expansion
Category: Math
Last updated: 2026-04-08