Examples & Applications

Step-by-step worked examples demonstrating the binomial theorem in action. From basic expansions to coefficient extraction and probability applications.

📝 Example 1: Expand (2x + 3y)³

Expanding a binomial with numerical coefficients requires applying the binomial theorem and then simplifying.

Step-by-Step Solution

Step 1: Identify coefficients from Pascal's triangle (row 3): 1, 3, 3, 1

Step 2: Apply to (2x + 3y)³:

(2x + 3y)³ = 1·(2x)³ + 3·(2x)²(3y) + 3·(2x)(3y)² + 1·(3y)³

Step 3: Compute each term:

= 1·8x³ + 3·4x²·3y + 3·2x·9y² + 1·27y³

Step 4: Simplify:

= 8x³ + 36x²y + 54xy² + 27y³

Visual representation of binomial expansion for (x+y)^4
Visualization of binomial expansion terms

✅ Verification: Set x=1, y=1 → (2+3)³ = 125; 8+36+54+27 = 125 ✓

🎯 Example 2: Coefficient Extraction

Finding the coefficient of a specific power in a binomial expansion is a common problem.

Problem

Find the coefficient of x⁷ in (2x − 3)¹⁰

Solution

Step 1: Rewrite as (2x + (−3))¹⁰

Step 2: General term formula:

Tk+1 = C(10,k) · (2x)¹⁰⁻ᵏ · (−3)ᵏ

Step 3: Find k where exponent of x is 7:

10 − k = 7 → k = 3

Step 4: Compute the coefficient:

Coefficient = C(10,3) · 2⁷ · (−3)³

= 120 · 128 · (−27)

= −414,720

💡 Key Technique

For (ax + b)ⁿ, the term containing xᵏ is Tn−k+1 = C(n, n−k) aᵏ bⁿ⁻ᵏ. Remember to include both the binomial coefficient AND the powers of a and b!

🎲 Example 3: Binomial Probability

The binomial theorem is the foundation of the binomial distribution in probability theory.

Problem

A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads?

Solution

Parameters: n = 5 trials, p = 0.5 (heads), k = 3 successes

Binomial PMF formula:

P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ

Substitute values:

P(X = 3) = C(5,3) × (0.5)³ × (0.5)²

= 10 × 0.125 × 0.25

= 10 × 0.03125

= 0.3125 (or 31.25%)

Binomial distribution probability mass function showing probabilities for n=5, p=0.5
Figure: Binomial distribution for n=5, p=0.5

∞ Example 4: Infinite Series (Generalized Binomial Theorem)

When the exponent is not a positive integer, we get an infinite series that converges for |x| < 1.

Problem

Find the first 4 terms of the expansion of (1 + x)¹ᐟ² (square root)

Solution

Generalized binomial coefficient:

C(½, k) = (½)(½−1)(½−2)...(½−k+1) / k!

Computing first 4 terms:

C(½, 0) = 1 → term: 1

C(½, 1) = ½ → term: (½)x

C(½, 2) = (½)(−½)/2 = −1/8 → term: −(1/8)x²

C(½, 3) = (½)(−½)(−3/2)/6 = 1/16 → term: +(1/16)x³

Result:

√(1+x) = 1 + ½x − ⅛x² + ₁₁₆x³ + ...

⚠️ Convergence Note

This infinite series converges only when |x| < 1. For |x| ≥ 1, the series diverges and does not equal the actual value.

✏️ Practice Problems

Problem 1: Expand (x + 2y)⁵

Solution:

Coefficients: 1, 5, 10, 10, 5, 1

= x⁵ + 5x⁴(2y) + 10x³(2y)² + 10x²(2y)³ + 5x(2y)⁴ + (2y)⁵

= x⁵ + 10x⁴y + 40x³y² + 80x²y³ + 80xy⁴ + 32y⁵

Problem 2: Sum of Coefficients Identity

Show that Σ C(n,k) = 2ⁿ

Proof: Set x = 1, y = 1 in the binomial theorem:

(1 + 1)ⁿ = Σ C(n,k) × 1ⁿ⁻ᵏ × 1ᵏ = Σ C(n,k)

2ⁿ = Σ C(n,k) ✓

Problem 3: Alternating Sum

Show that Σ (−1)ᵏ C(n,k) = 0 for n > 0

Proof: Set x = 1, y = −1:

(1 + (−1))ⁿ = Σ C(n,k) × 1ⁿ⁻ᵏ × (−1)ᵏ

0ⁿ = Σ (−1)ᵏ C(n,k)

0 = Σ (−1)ᵏ C(n,k) for n > 0 ✓

📋 Quick Reference: Common Expansions

(x + y)²

x² + 2xy + y²

(x + y)³

x³ + 3x²y + 3xy² + y³

(x + y)⁴

x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

(x + y)⁵

x⁵ + 5x⁴y + 10x³y² + 10x²y³ + 5xy⁴ + y⁵

🔗 Related Topics

📐 Definition

Learn the formal definition and properties of binomial coefficients.

🔬 Proofs

Understand why the binomial theorem is true through multiple proofs.

🖩 Calculator

Compute expansions instantly with our interactive calculator.

📚 Resources

References, textbooks, and LaTeX snippets for further study.