Permutation with Restriction
Permutation with Restriction problems involve arranging objects with specific constraints such as: certain objects must be together, must be apart, must be at fixed positions, or must be at ends. These problems require treating restricted groups as units or using complementary counting.
What You'll Learn
Introduction to Permutation with Restriction
Permutation with Restriction problems involve arranging objects with specific constraints such as: certain objects must be together, must be apart, must be at fixed positions, or must be at ends. These problems require treating restricted groups as units or using complementary counting.
Prerequisites
How to Solve Permutation with Restriction Problems
Step 1: Identify the type of restriction (together, apart, fixed position, ends)
Step 2: For 'together' problems: treat the group as a single unit
Step 3: Arrange the units (including the group unit) using permutation formula
Step 4: Multiply by internal arrangements within the group
Step 5: For 'apart' problems: use complementary counting or gap method
Step 6: For fixed position: fix that position first, then arrange remaining
Step 7: For 'ends' problems: handle end positions first, then arrange middle
Example Problem
Example: In how many ways can 5 people be seated in a row such that two specific people always sit together? Solution: Step 1: Treat the two specific people as a single unit Step 2: Now we have 4 units to arrange (the pair + 3 other individuals) Step 3: 4 units can be arranged in 4! = 24 ways Step 4: The two people within the pair can be arranged in 2! = 2 ways Step 5: Total = 24 × 2 = 48 ways Answer: 48 ways
Pro Tips & Tricks
- Together = treat as one unit, then multiply by internal arrangements
- Apart = total - together (complementary counting)
- Gap method for 'no two together': arrange unrestricted items first, then place restricted in gaps
- For 'ends' constraints: fill end positions first
- For specific positions: fix those positions, then arrange the rest
- When multiple groups must be together, treat each group as a separate unit
Shortcut Methods to Solve Faster
Common Mistakes to Avoid
Practice Worksheets
Practice makes perfect! Work through these worksheets to master Permutation with Restriction. Each worksheet contains 20 questions with detailed explanations. Start from Worksheet 1 and progress through increasing difficulty levels.
Exam Importance
Permutation with Restriction is an important topic for various competitive exams. Here's how frequently it appears:
Ready to Master Permutation with Restriction?
Start with Worksheet 1 and work your way up to expert level! Each worksheet includes: