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.