binomial theorem calculator

Expand (x + y)ⁿ, compute coefficients, and evaluate (a + b)ⁿ.

Exponent (n)

Term index (k, optional)

First symbol

Second symbol

Numeric value for a

Numeric value for b

Enter values and click Calculate to see the expansion and results.

If you are studying algebra, probability, or preparing for standardized tests, this binomial theorem calculator gives you the full expansion, individual term details, and numeric evaluation in one place. It is designed to be quick, readable, and useful for homework checks.

What is the binomial theorem?

The binomial theorem describes how to expand an expression of the form (x + y)ⁿ where n is a non-negative integer. Instead of multiplying the binomial repeatedly by hand, you can jump straight to the expanded form.

(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k

Here, C(n, k) or n choose k gives the coefficient for each term:

\binom{n}{k} = \frac{n!}{k!(n-k)!}

How to use this calculator

1) Enter the exponent n

Set the exponent to any whole number from 0 to 300. For small values, you will see the full expansion. For large values, the tool shows the leading and trailing terms for readability.

2) (Optional) Enter term index k

If you want a specific term, input k. The calculator returns the coefficient C(n,k) and the exact symbolic term C(n,k) x^n-k y^k.

3) Set symbols and numeric values

You can rename x and y to any letters. If you also provide numbers a and b, the tool evaluates (a+b)ⁿ so you get both symbolic and numeric results.

Why coefficients matter

The coefficients in a binomial expansion follow Pascal's Triangle. This pattern appears in combinatorics, probability distributions, polynomial approximations, and financial models that rely on repeated binary outcomes.

They count combinations.
They define each term's weight in the expansion.
They connect algebra with probability (e.g., binomial distribution).

Quick examples

Example A: Expand (x + y)⁵

Coefficients are 1, 5, 10, 10, 5, 1, so:

(x + y) 5 = x 5 + 5x 4 y + 10x 3 y 2 + 10x 2 y 3 + 5xy 4 + y 5

Example B: Evaluate (2 + 3)⁴

(2 + 3)⁴ = 5⁴ = 625. The calculator returns this directly and also shows each binomial term in the sum.

Common mistakes to avoid

Using a negative or non-integer exponent when applying the basic binomial theorem formula taught in introductory algebra.
Forgetting that k starts at 0 and ends at n.
Mixing up exponents: the first variable has power n-k, the second has power k.
Dropping coefficients from middle terms.

FAQ

Can this calculator handle very large n?

Yes, coefficients are computed with exact integer arithmetic. For very large expansions, output is shortened for readability.

Do I need to provide a and b?

No. If you only want symbolic expansion, leave numeric values as-is or ignore the numeric section.

Is this useful for probability classes?

Absolutely. Binomial coefficients are core to binomial probability, combinations, and counting methods.