Binomial Theorem Calculator

Work out binomial theorem instantly with clear inputs, formula shown and shareable results.

About Binomial Theorem Calculator

The binomial theorem expands (a + b)ⁿ as the sum over k of C(n, k)·a^(n−k)·bᵏ. There are n + 1 terms, and the coefficients form row n of Pascal's triangle.

Formula

Binomial theorem

(a + b)ⁿ = Σₖ₌₀ⁿ C(n, k)·a^(n−k)·bᵏ

Worked example

With the default values below, Binomial Theorem Calculator returns Value of the chosen term of 720.000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

First term a in (a + b)ⁿ: 2
Second term b: 3
Power naccepts 0–60: 5
Which term to isolate (k, from 0)accepts 0–60: 2

Results

Value of the chosen term: 720.000000
C(n, k): 10
(a + b)ⁿ in total: 3,125.000000
Number of terms in the expansion: 6

Step by step

1Values used
2Binomial theorem
3Value of the chosen term= 720.000000
4C(n, k)= 10
5(a + b)ⁿ in total= 3,125.000000
6Number of terms in the expansion= 6

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

The k = 2 term of (2 + 3)⁵?

C(5,2)·2³·3² = 10 × 8 × 9 = 720, out of a total of 5⁵ = 3125.

How do I find the middle term?

For even n it is k = n/2; for odd n there are two middle terms at k = (n−1)/2 and (n+1)/2.