Mathematical Explorer
Binomial
Theorem
(a + b)n = ∑k=0n C(n,k) · an−k · bk
Expand, visualize, and explore binomial expressions interactively.
01 — Calculator
Expand Any Binomial
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Step-by-Step Expansion
02 — Coefficients
Binomial Coefficients
03 — Pascal's Triangle
Pascal's Triangle
Each number is the sum of the two directly above it. The n-th row gives the coefficients for (a+b)n.
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04 — Theory
The Theorem Explained
General Formula
For any real numbers a and b, and non-negative integer n:
(a + b)n = Σ C(n,k) · an−k · bk
where k ranges from 0 to n.
Binomial Coefficient
C(n, k) — read as "n choose k" — counts the number of ways to choose k items from n:
C(n,k) = n! / (k! · (n−k)!)
Special Cases
Some elegant identities that follow from the theorem:
2n = Σ C(n,k) [a=b=1]
0 = Σ (−1)kC(n,k) [a=1,b=−1]
Properties
- C(n,0) = C(n,n) = 1
- C(n,k) = C(n, n−k) (symmetry)
- C(n,k) + C(n,k+1) = C(n+1,k+1)
- Sum of row n = 2n