Binomial Theorem Definition — Formula, Coefficients & Pascal's Triangle

📐 Formal Statement

For any nonnegative integer n and any real or complex numbers x, y:

(x + y)ⁿ = Σ_k=0ⁿ (ⁿ/_k) x^n−k y^k

Where the binomial coefficient (ⁿ/_k) (read "n choose k") equals:

(ⁿ/_k) = n! / (k! × (n−k)!)

Historical Note: This formula was first stated by Euclid around 300 BCE for n=2, and generalized by Brahmagupta in the 7th century. The modern notation was introduced by Christoph Rudolff in 1525.

🔢 Binomial Coefficients

The binomial coefficient (ⁿ/_k) counts the number of ways to choose k items from a set of n distinct items without regard to order.

Computation Methods

Factorial form: C(n,k) = n! / (k!(n−k)!)
Multiplicative form: C(n,k) = n×(n−1)×...×(n−k+1) / k!
Recursive form: C(n,k) = C(n−1,k) + C(n−1,k−1)

Key Properties

Symmetry: C(n,k) = C(n, n−k)
Boundary: C(n,0) = C(n,n) = 1
Pascal's Identity: C(n+1,k) = C(n,k) + C(n,k−1)

📊 Combination Visualization

Visual representation of combination C(n,k) — selecting k items from n without regard to order — Figure 1: Visual representation of combination C(n,k)

For example, C(4,2) = 6 ways to choose 2 items from 4.

🔺 Pascal's Triangle

Binomial coefficients form a triangular array known as Pascal's Triangle. Each entry is the sum of the two entries directly above it.

How to Read the Triangle

Row n contains the coefficients for (x+y)ⁿ
The first and last entries are always 1
Each interior entry equals the sum of the two above it
The triangle is symmetric

📈 Key Identities

These identities are derived from the binomial theorem and are essential in combinatorics:

Sum of All Coefficients

Σ_k=0ⁿ C(n,k) = 2ⁿ

Number of subsets of an n-element set.

Alternating Sum

Σ (−1)^k C(n,k) = 0 (n > 0)

Equal numbers of even and odd-sized subsets.

Weighted Sum

Σ k × C(n,k) = n × 2ⁿ⁻¹

Expected sum of all subset sizes.

🎲 Binomial Distribution

Binomial coefficients form the basis of the binomial distribution, which models the number of successes in n independent Bernoulli trials.

Probability Mass Function

P(X = k) = C(n,k) × p^k × (1−p)^n−k

Example: P(exactly 3 heads in 5 flips) = C(5,3) × 0.5³ × 0.5² = 10 × 0.125 = 0.125 (12.5%)

📝 Example: (x + y)⁴

Using row 4 of Pascal's triangle: 1, 4, 6, 4, 1

(x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

Visual expansion of (x+y)^4 — Figure 4: Visual expansion of (x+y)⁴

✅ Verification: x=1, y=1 → 1+4+6+4+1 = 16 = 2⁴

💻 Computation Algorithms

Efficient computation of binomial coefficients for large n:

Multiplicative Algorithm (avoids large factorials)

function binomialCoefficient(n, k):
    if k < 0 or k > n: return 0
    if k > n - k: k = n - k
    result = 1
    for i in range(1, k + 1):
        result = result * (n - i + 1) / i
    return result

Generate Pascal's Row

row = [1]
for k in 1..n:
    row[k] = row[k-1] * (n - k + 1) / k

💡 Pro Tips

Use symmetry: C(n,k) = C(n, n−k)
Python: math.comb(n, k) for arbitrary precision
JavaScript: Use BigInt for large coefficients

Definition & Formal Statement