Combination with 'At Least' Constraint
Combination with 'At Least' Constraint problems require selecting a committee or group where certain categories must have at least a minimum number of members (e.g., at least 2 men, at least 1 woman). These are solved by summing combinations for each valid case or using complementary counting.
What You'll Learn
Introduction to Combination with 'At Least' Constraint
Combination with 'At Least' Constraint problems require selecting a committee or group where certain categories must have at least a minimum number of members (e.g., at least 2 men, at least 1 woman). These are solved by summing combinations for each valid case or using complementary counting.
Prerequisites
How to Solve Combination with 'At Least' Constraint Problems
Step 1: Identify the minimum requirements for each category
Step 2: List all possible distributions that satisfy the requirements
Step 3: For each valid case, calculate number of ways using combinations
Step 4: Sum the results (Sum Rule - cases are mutually exclusive)
Step 5: Alternatively, use complementary counting: total - undesirable cases
Step 6: Verify that the sum of selections equals total committee size
Step 7: Present the final answer
Example Problem
Example: A committee of 5 is to be formed from 6 men and 4 women, with at least 2 men. How many ways? Solution: Step 1: At least 2 men means 2, 3, 4, 5 men (but max men = 6, max women = 4) Step 2: Cases: - 2 men, 3 women: ⁶C₂ × ⁴C₃ = 15 × 4 = 60 - 3 men, 2 women: ⁶C₃ × ⁴C₂ = 20 × 6 = 120 - 4 men, 1 woman: ⁶C₄ × ⁴C₁ = 15 × 4 = 60 - 5 men, 0 women: ⁶C₅ × ⁴C₀ = 6 × 1 = 6 Step 3: Total = 60 + 120 + 60 + 6 = 246 Answer: 246 ways
Pro Tips & Tricks
- At least k = sum from k to maximum
- At most k = sum from 0 to k
- Use complementary counting when 'at least' covers most cases
- For 'at least 1' from a group: total - selections with none from that group
- Always ensure cases don't exceed available members
- List all possible distributions systematically
Shortcut Methods to Solve Faster
Common Mistakes to Avoid
Practice Worksheets
Practice makes perfect! Work through these worksheets to master Combination with 'At Least' Constraint. Each worksheet contains 20 questions with detailed explanations. Start from Worksheet 1 and progress through increasing difficulty levels.
Exam Importance
Combination with 'At Least' Constraint is an important topic for various competitive exams. Here's how frequently it appears:
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