Explanation

The binomial theorem expands (a + b)ⁿ without multiplying manually.

Formula:
(a + b)ⁿ = Σ (k=0 to n) [nCk * aⁿ⁻ᵏ * bᵏ]

Where:

• nCk = combination = n! / (k!(n−k)!)

• k = term number (starts at 0)

Example:
(a + b)³ = 3C0 a³b⁰ + 3C1 a²b¹ + 3C2 a¹b² + 3C3 a⁰b³ = a³ + 3a²b + 3ab² + b³

Quiz

1. Expand (x + y)² using the binomial theorem.

2. What is nCk?

3. Expand (a + b)³.

4. How many terms in (x + y)⁵?

5. Is the binomial theorem useful for large powers?

Answer Key

1. x² + 2xy + y²

2. Combination n! / (k!(n−k)!)

3. a³ + 3a²b + 3ab² + b³

4. 6 terms (n+1)

5. Yes