What is the binomial theorem formula?

The binomial theorem states that (a + b)^n equals the sum from k = 0 to n of C(n,k) times a^(n-k) times b^k. Each term has a binomial coefficient C(n,k) = n! / (k!(n-k)!), a descending power of a, and an ascending power of b.

How many terms are in a binomial expansion?

An expansion of (a + b)^n has exactly n + 1 terms. For example, (a + b)^2 has 3 terms, (a + b)^7 has 8 terms, and (a + b)^10 has 11 terms. The number of terms equals the number of entries in row n of Pascal's triangle.

How do you find a specific term without expanding the whole thing?

Use the general term formula T(k+1) = C(n,k) times a^(n-k) times b^k. For the (k+1)th term, set k accordingly, compute the coefficient C(n,k), then attach a^(n-k) and b^k. For example, the 3rd term of (x + 2)^5 uses k = 2: C(5,2) times x^3 times 2^2 = 40x^3.

Can the binomial theorem be used with a negative or fractional exponent?

Yes, but not with the simple finite formula. For a negative or fractional exponent you use the generalized binomial series, which is an infinite sum that converges only for certain values. This calculator handles non-negative integer powers (n from 0 to 10), which covers standard algebra and pre-calculus problems.

What is the difference between the binomial theorem and the binomial distribution?

The binomial theorem is an algebra tool for expanding (a + b)^n into a polynomial. The binomial distribution is a probability model for the number of successes in n independent trials. They share the coefficient C(n,k) but answer different questions: algebra versus probability.

Binomial Theorem Calculator

Formula

Enter your values

Enter your values and press Calculate — your answer, visual and step-by-step solution appear here.

This binomial theorem calculator expands (a + b)ⁿ and shows every coefficient, term, and Pascal's-triangle row — so you can both get the answer and see exactly how it was reached.

What Is the Binomial Theorem?

The binomial theorem is a formula that expands a binomial — an expression with two terms, like (a + b) — raised to a whole-number power n, without multiplying it out by hand n times. Instead of laboriously computing (a + b)(a + b)(a + b)…, the theorem tells you every term of the result at once.

In plain English, it is a shortcut. The expansion of (a + b)ⁿ always has exactly n + 1 terms, the powers of a count down from n to 0, the powers of b count up from 0 to n, and each term carries a number called a binomial coefficient. The binomial theorem calculator above applies this rule instantly for any n from 0 to 10.

Key fact: Expanding (a + b)ⁿ produces n + 1 terms. So (a + b)³ has 4 terms, (a + b)⁵ has 6 terms, and (a + b)¹⁰ has 11 terms.

Binomial Theorem Formula (Explained)

The binomial theorem is written compactly using summation notation:

(a + b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ (k = 0 → n)

Each part of the formula means something specific:

n — the power the binomial is raised to (a non-negative integer).
k — the term index, running from 0 up to n; it labels each of the n + 1 terms.
C(n,k) — the binomial coefficient, read "n choose k." It is exactly a combination (n choose k) and gives the number in front of each term.
aⁿ⁻ᵏ — the first term raised to a descending power (n, n−1, n−2, …, 0).
bᵏ — the second term raised to an ascending power (0, 1, 2, …, n).

The binomial coefficient itself is computed with factorials:

C(n,k) = n! / [ k! (n − k)! ]

This calculator uses exact integer factorials and coefficients, so there is no floating-point rounding error in the numbers it returns.

How to Expand a Binomial Step by Step

You can reproduce the calculator's result by hand in five steps. This is the method graders expect to see on an exam:

Identify a, b, and n. Read off the first term (a), the second term (b, including its sign), and the power (n).
Write the n + 1 coefficients. Take row n of Pascal's triangle, or compute C(n,0), C(n,1), …, C(n,n) directly.
Write the descending powers of a. Attach aⁿ, aⁿ⁻¹, aⁿ⁻², …, a⁰ to the coefficients in order. Use the exponent calculator if the powers of a get messy.
Write the ascending powers of b. Attach b⁰, b¹, b², …, bⁿ to the same terms, so each term's exponents add up to n.
Multiply and combine. Simplify each term (coefficient × power of a × power of b) and write them in a single line.

Or let the binomial theorem calculator do all five steps instantly and show the working.

Worked Examples

Example 1 — Expand (x + 3)²

Here a = x, b = 3, and n = 2. Row 2 of Pascal's triangle is 1, 2, 1, so there are n + 1 = 3 terms.

Term 1: 1 · x² · 3⁰ = x²
Term 2: 2 · x¹ · 3¹ = 6x
Term 3: 1 · x⁰ · 3² = 9

Combining gives (x + 3)² = x² + 6x + 9. This is the perfect-square pattern, and it is also the first step of expanding many quadratic expressions.

Example 2 — Expand (2x − 3)⁴

Here a = 2x, b = −3, and n = 4. Row 4 of Pascal's triangle is 1, 4, 6, 4, 1. Because b is negative, the signs alternate + − + − +.

Term 1: 1 · (2x)⁴ · (−3)⁰ = 16x⁴
Term 2: 4 · (2x)³ · (−3)¹ = 4 · 8x³ · (−3) = −96x³
Term 3: 6 · (2x)² · (−3)² = 6 · 4x² · 9 = 216x²
Term 4: 4 · (2x)¹ · (−3)³ = 4 · 2x · (−27) = −216x
Term 5: 1 · (2x)⁰ · (−3)⁴ = 81

So (2x − 3)⁴ = 16x⁴ − 96x³ + 216x² − 216x + 81. Keep the entire second term, including its sign, inside the powers: it is (−3)², not −3², which would give the wrong sign.

Example 3 — Find a single term without expanding

To pull out one term without writing the whole expansion, use the general term formula for the (k + 1)th term:

T(k+1) = C(n,k) aⁿ⁻ᵏ bᵏ

Find the 3rd term of (x + 2)⁵. The 3rd term means k + 1 = 3, so k = 2, with a = x, b = 2, n = 5.

Set k from the term number. The 3rd term ⇒ k + 1 = 3 ⇒ k = 2.
Find the coefficient. C(5, 2) = 5! / (2! · 3!) = 10.
Apply the powers. aⁿ⁻ᵏ = x⁵⁻² = x³ and bᵏ = 2² = 4.
Multiply. T₃ = 10 · x³ · 4 = 40x³.

The 3rd term of (x + 2)⁵ is 40x³ — found directly, with no full expansion needed.

Pascal's Triangle and the Binomial Coefficients

Pascal's triangle is a fast way to read off the binomial coefficients. Row n of the triangle gives the coefficients C(n,0), C(n,1), …, C(n,n) — exactly the numbers in front of each term of (a + b)ⁿ. In the calculator above, this is the highlighted row shown with each expansion.

The triangle has three properties worth knowing: each entry is the sum of the two entries directly above it; every row is symmetric (it reads the same left to right as right to left, because C(n,k) = C(n, n−k)); and the entries of row n always sum to 2ⁿ. Each row is also a short numeric sequence — explore more patterns with the sequence calculator.

n	Row of Pascal's triangle (coefficients)	Sum (2ⁿ)
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

Binomial Theorem vs. Binomial Distribution

These two ideas are often confused because they share the binomial coefficient C(n,k), but they answer completely different questions. The binomial theorem is an algebra tool: it expands (a + b)ⁿ into a sum of terms — which is what this page does. The binomial distribution is a probability model: it gives the chance of getting exactly k successes in n independent trials.

So if you want to rewrite an expression like (2x − 3)⁴ as a polynomial, you need the theorem (this calculator). If you want the probability of, say, 3 heads in 5 coin flips, you need the distribution instead. Same coefficient, different jobs.

Common Mistakes to Avoid

Mistake 1 — Expecting n terms instead of n + 1An expansion of (a + b)ⁿ always has n + 1 terms. (a + b)⁴ has five terms, not four. Count the coefficients in row n of Pascal's triangle to check.

Mistake 2 — Sign errors when the second term is negativeWhen b is negative, keep the sign inside the power: (−3)² = 9, but −3² = −9. The signs of the terms then alternate + − + −. This is the single most common slip.

Mistake 3 — Mixing up ascending and descending powersThe power of a goes down (n → 0) while the power of b goes up (0 → n). In every term the two exponents must add to n; if they don't, a power is wrong.

Mistake 4 — Misreading the coefficient C(n,k)C(n,k) = n! / [k!(n − k)!], not n/k. For example C(5,2) = 10, not 2.5. Read the coefficient straight from row n of Pascal's triangle to avoid arithmetic errors.

Mistake 5 — Using the plain formula for fractional or negative nThe finite formula here works only for non-negative integer n. For a fractional or negative exponent you need the generalized binomial series, which is an infinite sum. This calculator handles integer n from 0 to 10.

Studying for AP Statistics? The same C(n,k) coefficient is the binomial probability formula in Unit 2.

Frequently Asked Questions

What is the binomial theorem formula?

The binomial theorem states that (a + b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ for k = 0 to n. Each term has a binomial coefficient C(n,k) = n! / [k!(n − k)!], a descending power of a, and an ascending power of b. The binomial theorem calculator above evaluates this for any n from 0 to 10.
How many terms are in a binomial expansion?

An expansion of (a + b)ⁿ has exactly n + 1 terms. For example, (a + b)² has 3 terms, (a + b)⁷ has 8 terms, and (a + b)¹⁰ has 11 terms. The number of terms equals the number of entries in row n of Pascal's triangle.
How do you find a specific term without expanding the whole thing?

Use the general term formula T(k+1) = C(n,k) aⁿ⁻ᵏ bᵏ. For the (k+1)th term, set k accordingly, compute the coefficient C(n,k), then attach aⁿ⁻ᵏ and bᵏ. For instance, the 3rd term of (x + 2)⁵ uses k = 2: C(5,2) · x³ · 2² = 40x³.
Can the binomial theorem be used with a negative or fractional exponent?

Yes, but not with the simple finite formula. For a negative or fractional exponent you use the generalized binomial series, which is an infinite sum that converges only for certain values. This calculator handles non-negative integer powers (n from 0 to 10), which covers standard algebra and pre-calculus problems.
What is the difference between the binomial theorem and the binomial distribution?

The binomial theorem is an algebra tool for expanding (a + b)ⁿ into a polynomial. The binomial distribution is a probability model for the number of successes in n independent trials. They share the coefficient C(n,k) but answer different questions — algebra versus probability.

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