Binomial Theorem Expansion Calculator
Expand (a + b)^n step by step using the binomial theorem.
Shows each term, binomial coefficients, and Pascal's triangle row for any power up to 20.
The Binomial Theorem
The binomial theorem gives a formula for expanding any power of a two-term expression (a + b)^n without multiplying it out term by term.
The formula:
(a + b)^n = sum from k=0 to n of C(n, k) * a^(n-k) * b^k
Where C(n, k) = n! / (k! * (n-k)!) is the binomial coefficient, read as “n choose k.”
Pascal’s Triangle connection: Each row of Pascal’s triangle gives the binomial coefficients for a given power n:
| n | Coefficients |
|---|---|
| 0 | 1 |
| 1 | 1 1 |
| 2 | 1 2 1 |
| 3 | 1 3 3 1 |
| 4 | 1 4 6 4 1 |
| 5 | 1 5 10 10 5 1 |
Each number is the sum of the two numbers directly above it.
Worked example — (2x + 3)^3:
- k=0: C(3,0) * (2x)^3 * 3^0 = 1 * 8x^3 * 1 = 8x^3
- k=1: C(3,1) * (2x)^2 * 3^1 = 3 * 4x^2 * 3 = 36x^2
- k=2: C(3,2) * (2x)^1 * 3^2 = 3 * 2x * 9 = 54x
- k=3: C(3,3) * (2x)^0 * 3^3 = 1 * 1 * 27 = 27
- Result: (2x + 3)^3 = 8x^3 + 36x^2 + 54x + 27
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 (substitute a=1, b=1)
- Coefficients grow quickly — C(20,10) = 184,756
Where it is used: The binomial theorem appears in probability theory (binomial distributions), calculus (approximating (1+x)^n for small x), combinatorics, statistics (confidence intervals), and even in physics for perturbation expansions.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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