Binomial Expansion
Expand (a+b)ⁿ and show its terms
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Binomial Expansion turns the numbers behind expanding (a+b)ⁿ and show its terms into an instant, step-by-step result. You supply a, b and n (power) to Binomial Expansion; the answer follows at once. Binomial Expansion displays each intermediate figure as well. Binomial Expansion shines when speed matters. No account is needed for Binomial Expansion. The browser does the arithmetic. Binomial Expansion is part of the Math collection on CalcProMaster, alongside binomial expansion calculator coefficient, free online binomial expansion calculator and more. Keep Binomial Expansion bookmarked for next time. Binomial Expansion stays bookmarked for next time.
What the defaults give
(2 + 3)⁵ = 3,125, and the coefficients are 1, 5, 10, 10, 5, 1.
That is 2⁵ + 5×2⁴×3 + 10×2³×3² + 10×2²×3³ + 5×2×3⁴ + 3⁵, which adds to 3,125.
The theorem
(a + b)ⁿ = Σ C(n,k) × aⁿ⁻ᵏ × bᵏ, for k = 0 to n
C(n,k) = n! ÷ (k! × (n−k)!)
The coefficient C(n,k) counts how many ways k factors of b can be chosen from n brackets. That is why the row for n = 5 is 1, 5, 10, 10, 5, 1: there are five ways to pick one b, ten ways to pick two, and so on.
Worked examples
(2 + 3)⁵ → coefficients 1, 5, 10, 10, 5, 1 → 3,125
(1 + 1)⁵ → 32, and the coefficients sum to 32
(1 + 2)³ → 1 + 6 + 12 + 8 = 27
(2 + 1)⁴ → 16 + 32 + 24 + 8 + 1 = 81
(a + b)¹⁰ → eleven terms, coefficients 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1
(x − 2)³ → x³ − 6x² + 12x − 8, using b = −2
Pascal triangle and the check that catches errors
Each coefficient row is the sum of the two above it, which is why the triangle is easier to write than the factorial formula for small n. The row always starts and ends with 1, is symmetric, and has n + 1 entries. A quick sanity check on any expansion: the coefficients must sum to 2ⁿ, and the whole expression evaluated at a = b = 1 must equal 2ⁿ. For n = 5 that is 32, and the row 1, 5, 10, 10, 5, 1 does sum to 32.
The sign pattern is the other trap. In (a − b)ⁿ the terms alternate, because a negative b raised to an odd power stays negative. Writing the signs as part of b avoids most mistakes: treat the second term as −2 rather than trying to remember whether the cube term is plus or minus.
Where it is used
In probability, (p + q)ⁿ with p + q = 1 gives the binomial distribution directly — the coefficient counts arrangements and the powers give the probability of each outcome. In finance, the same expansion underlies compound interest approximations and the binomial option pricing model, where a price moves up or down at each step.
In algebra it is the shortcut for expanding powers without multiplying brackets repeatedly, and the first two terms give the standard approximation (1 + x)ⁿ ≈ 1 + nx for small x, which is accurate enough in physics and engineering when x is below about 0.1. The binomial theorem is what justifies that approximation and tells you the size of the error you are accepting.
Frequently Asked Questions
What are the coefficients for (a+b)⁵?
1, 5, 10, 10, 5, 1. They come from C(5,k), and they sum to 32, which is 2⁵.
How do I expand (2+3)⁵?
Use the coefficients with descending powers of 2 and ascending powers of 3: 32 + 240 + 720 + 1080 + 810 + 243, which totals 3,125.
Why does the triangle work?
Each entry is the sum of the two above it, which follows from Pascal identity C(n,k) = C(n−1,k−1) + C(n−1,k). It is faster for small n than factorials.
How do I handle a minus sign?
Treat the second term as negative, so (x − 2)³ uses b = −2 and the signs alternate: x³ − 6x² + 12x − 8.
What is a quick check on an expansion?
The coefficients must add to 2ⁿ. For n = 5 they sum to 32; if your row sums to anything else, a coefficient is wrong.
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