Partition into Unequal Groups
Partition into Unequal Groups involves dividing a set of distinct items into groups where the groups have different sizes. When group sizes are all different, the groups are naturally distinguishable by size, so no division by factorial is needed. This differs from equal groups where division by k! is required.
What You'll Learn
Introduction to Partition into Unequal Groups
Partition into Unequal Groups involves dividing a set of distinct items into groups where the groups have different sizes. When group sizes are all different, the groups are naturally distinguishable by size, so no division by factorial is needed. This differs from equal groups where division by k! is required.
Prerequisites
How to Solve Partition into Unequal Groups Problems
Step 1: Identify total items (n) and group sizes (n₁, n₂, ..., nₖ)
Step 2: Check if all group sizes are different (unequal)
Step 3: If sizes are all different, groups are distinguishable by size
Step 4: Calculate using multiplication of combinations: C(n, n₁) × C(n-n₁, n₂) × ...
Step 5: Do NOT divide by k! (unlike equal groups)
Step 6: If some groups have equal size, divide by factorial of those equal groups
Step 7: Verify that sum of group sizes equals n
Example Problem
Example: Divide 10 people into groups of sizes 3, 3, and 4. How many ways? (Groups unlabeled but sizes 3,3,4) Solution: Step 1: n = 10, sizes: 3, 3, 4 (two groups of size 3 are equal) Step 2: Groups are unlabeled, but two groups of size 3 are indistinguishable Step 3: First, calculate as if groups were labeled: C(10,3) × C(7,3) × C(4,4) = 120 × 35 × 1 = 4200 Step 4: Since two groups of size 3 are indistinguishable, divide by 2! = 2 Step 5: Total = 4200 / 2 = 2100 Answer: 2100 ways
Pro Tips & Tricks
- All group sizes different: groups are automatically distinguishable
- Equal group sizes: divide by k! (where k = number of equal groups)
- For multiple groups of same size, divide by factorial for each size group
- The order of selecting groups doesn't matter when groups are unlabeled
- Use multinomial coefficient: n!/(n₁! n₂! ... nₖ!) × (1/ (k₁! k₂! ...)) for unlabeled groups
- When groups are labeled, don't divide by any factorial
Shortcut Methods to Solve Faster
Common Mistakes to Avoid
Practice Worksheets
Practice makes perfect! Work through these worksheets to master Partition into Unequal Groups. Each worksheet contains 20 questions with detailed explanations. Start from Worksheet 1 and progress through increasing difficulty levels.
Exam Importance
Partition into Unequal Groups is an important topic for various competitive exams. Here's how frequently it appears:
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Start with Worksheet 1 and work your way up to expert level! Each worksheet includes: