Multinomial Theorem
The Multinomial Theorem generalizes the Binomial Theorem to more than two terms. The coefficient of x₁ᵃ x₂ᵇ ... xₖᶜ in the expansion of (x₁ + x₂ + ... + xₖ)ⁿ is n!/(a! b! c! ...), where a + b + c + ... = n. This is also the number of ways to arrange n objects with a of type 1, b of type 2, etc.
What You'll Learn
Introduction to Multinomial Theorem
The Multinomial Theorem generalizes the Binomial Theorem to more than two terms. The coefficient of x₁ᵃ x₂ᵇ ... xₖᶜ in the expansion of (x₁ + x₂ + ... + xₖ)ⁿ is n!/(a! b! c! ...), where a + b + c + ... = n. This is also the number of ways to arrange n objects with a of type 1, b of type 2, etc.
Prerequisites
How to Solve Multinomial Theorem Problems
Step 1: Identify the expression (x₁ + x₂ + ... + xₖ)ⁿ
Step 2: Identify the exponents for each variable (a, b, c, ...) that sum to n
Step 3: Apply multinomial coefficient formula: n!/(a! b! c! ...)
Step 4: This coefficient is the number of times the term appears
Step 5: For coefficient of specific term, multiply by any constants in the expression
Step 6: For problems about number of ways to arrange, use the same formula
Step 7: Verify that exponents sum to n
Example Problem
Example: Find the coefficient of x²y³z in (x + y + z)⁶ Solution: Step 1: Exponents: x², y³, z¹ (2 + 3 + 1 = 6) ✓ Step 2: Formula: 6! / (2! × 3! × 1!) Step 3: 6! = 720, 2! = 2, 3! = 6, 1! = 1 Step 4: 720 / (2 × 6 × 1) = 720 / 12 = 60 Answer: 60
Pro Tips & Tricks
- Multinomial coefficient = n!/(a! b! c! ...)
- For 2 variables, reduces to binomial coefficient: C(n, a)
- Sum of all multinomial coefficients for given n = kⁿ
- The multinomial coefficient counts permutations with identical objects
- Can be calculated as: C(n, a) × C(n-a, b) × C(n-a-b, c) × ...
- Useful in probability (multinomial distribution)
Shortcut Methods to Solve Faster
Common Mistakes to Avoid
Practice Worksheets
Practice makes perfect! Work through these worksheets to master Multinomial Theorem. Each worksheet contains 20 questions with detailed explanations. Start from Worksheet 1 and progress through increasing difficulty levels.
Exam Importance
Multinomial Theorem is an important topic for various competitive exams. Here's how frequently it appears:
Ready to Master Multinomial Theorem?
Start with Worksheet 1 and work your way up to expert level! Each worksheet includes: