Binomial Expansion Practice- Problems & Solutions

What Is Binomial Expansion? Let's Cut to the Chase

Binomial expansion is just a way to expand expressions raised to a power. You know (a + b)² — that's a binomial. Multiply it out: a² + 2ab + b². That's expansion. Now do that for (a + b)¹⁰. Good luck doing it by hand.

The binomial theorem gives you a shortcut. It tells you exactly what each term looks like without multiplying everything out term by term. That's it. That's the whole point.

The Binomial Theorem Formula

Here's the formula you need:

(a + b)ⁿ = Σ C(n,k) · aⁿ⁻ᵏ · bᵏ

That Σ symbol means "sum from k = 0 to n". Each term follows the same pattern:

• C(n,k) is the binomial coefficient (also written as "n choose k")
• aⁿ⁻ᵏ is the first term with decreasing powers
• bᵏ is the second term with increasing powers

The coefficients follow a pattern. Write out the first few rows of Pascal's Triangle and you'll see them:

• Row 0: 1
• Row 1: 1 1
• Row 2: 1 2 1
• Row 3: 1 3 3 1
• Row 4: 1 4 6 4 1

How to Calculate Binomial Coefficients

You can calculate C(n,k) two ways:

Method 1: Factorials

C(n,k) = n! / (k! · (n-k)!)

Example: C(5,2) = 5! / (2! · 3!) = 120 / (2 · 6) = 120/12 = 10

Method 2: Pascal's Triangle

Find row n, go to position k. That's your coefficient. For C(5,2), look at row 5: 1 5 10 10 5 1. The third position (counting from 0) is 10.

Quick Comparison

Practice Problems with Solutions

Work through these. Try them before checking the answers.

Problem 1: Expand (x + 2)³

Step 1: Identify n = 3. Use row 3 of Pascal's Triangle: 1 3 3 1

Step 2: Apply the pattern. Terms go from k = 0 to k = 3.

• k = 0: C(3,0) · x³ · 2⁰ = 1 · x³ · 1 = x³
• k = 1: C(3,1) · x² · 2¹ = 3 · x² · 2 = 6x²
• k = 2: C(3,2) · x¹ · 2² = 3 · x · 4 = 12x
• k = 3: C(3,3) · x⁰ · 2³ = 1 · 1 · 8 = 8

Answer: (x + 2)³ = x³ + 6x² + 12x + 8

Problem 2: Expand (2x - 3)⁴

Step 1: n = 4. Row 4: 1 4 6 4 1

Step 2: Watch the signs. When one term is negative, alternate signs. For (a - b)ⁿ, signs go +, -, +, -, ...

• k = 0: C(4,0) · (2x)⁴ · (-3)⁰ = 1 · 16x⁴ · 1 = 16x⁴
• k = 1: C(4,1) · (2x)³ · (-3)¹ = 4 · 8x³ · (-3) = -96x³
• k = 2: C(4,2) · (2x)² · (-3)² = 6 · 4x² · 9 = 216x²
• k = 3: C(4,3) · (2x)¹ · (-3)³ = 4 · 2x · (-27) = -216x
• k = 4: C(4,4) · (2x)⁰ · (-3)⁴ = 1 · 1 · 81 = 81

Answer: (2x - 3)⁴ = 16x⁴ - 96x³ + 216x² - 216x + 81

Problem 3: Find the coefficient of x⁵ in (x + 3)⁸

Step 1: You don't need to expand the whole thing. Find which k gives x⁵.

General term: C(8,k) · x⁸⁻ᵏ · 3ᵏ

Step 2: Set exponent equal to 5: 8 - k = 5 → k = 3

Step 3: Plug in k = 3: C(8,3) · x⁵ · 3³

C(8,3) = 56 (from row 8: 1 8 28 56 70 56 28 8 1)

56 · 3³ = 56 · 27 = 1512

Answer: The coefficient is 1512

Problem 4: Expand (1 + x)⁶ - (1 + x)⁵

Step 1: Expand each separately using Pascal's Triangle.

Row 6: 1 6 15 20 15 6 1

Row 5: 1 5 10 10 5 1

Step 2: Write out both expansions and subtract.

(1 + x)⁶ = 1 + 6x + 15x² + 20x³ + 15x⁴ + 6x⁵ + x⁶

(1 + x)⁵ = 1 + 5x + 10x² + 10x³ + 5x⁴ + x⁵

Step 3: Subtract term by term.

Answer: x + 5x² + 10x³ + 10x⁴ + 5x⁵ + x⁶

Common Mistakes to Avoid

• Forgetting Pascal's Triangle: Always write out the coefficients first. Trying to calculate C(n,k) for each term mid-expansion is a waste of time and increases errors.
• Sign errors with (a - b)ⁿ: The signs alternate starting with positive. Don't just make the second term negative and forget the rest.
• Miscounting powers: The power of the first term decreases from n to 0. The power of the second term increases from 0 to n. Write them out explicitly until this becomes automatic.
• Confusing C(n,k) with C(k,n): C(n,k) means "n choose k". Order matters. C(5,2) = 10, but C(2,5) doesn't exist.
• Rushing the factorial arithmetic: Simplify factorials before multiplying. 10! / 8! = 10 × 9 = 90, not 10! / 8! = 3628800 / 40320.

How to Get Started: Your Practice Routine

Follow these steps in order. Don't skip ahead.

Step 1: Master Pascal's Triangle

Memorize rows 0 through 10. Write them out five times. Quiz yourself until you can reproduce any row instantly. This takes 10 minutes and saves hours later.

Step 2: Expand Five Binomials by Hand

Start with n = 2, 3, 4. Use (x + 1), (x + 2), (2x + 1). Write every coefficient. Check against your answers. Build the muscle memory.

Step 3: Add Negative Terms

Once you're comfortable with positives, practice (x - 1), (2x - 3), (1 - x)². The alternating signs trip people up. Drill them specifically.

Step 4: Find Specific Terms Only

Stop expanding the whole expression. Find the 4th term of (x + 2)⁷. Find the coefficient of x⁶ in (2x - 1)⁹. These appear on exams. Learn to identify k from the exponent you want.

Step 5: Mix It Up

Combine binomial expansion with differentiation or integration. Add constants. Multiply binomials by binomials. Real problems don't come in clean formats.

When Binomial Expansion Shows Up

You'll encounter this in:

• Algebra exams: Direct expansion and finding specific coefficients
• Probability: Binomial distribution uses the same coefficients
• Calculus: Taylor series are built on binomial concepts
• Approximations: (1 + x)ⁿ ≈ 1 + nx for small x (this is the binomial approximation)

Once you see the pattern, it's the same operation every time. The numbers change, the structure doesn't. Get the structure right and you can solve any binomial expansion problem.