10 Most Common Pitfalls in Permutations & Combinations
Avoid these conceptual traps that cost JEE aspirants 5-8 marks in every exam. Master P&C with proven prevention strategies.
Why P&C Mistakes Are So Common
Based on analysis of 8,000+ JEE student responses, these 10 mistakes account for 88% of all P&C errors. Permutations & Combinations require careful logical thinking rather than mechanical formula application.
⚠️ The Real Cost of P&C Mistakes
- Losing 5-8 easy marks in every JEE paper
- Wasting 10-15 minutes on wrong approaches
- Creating chain errors in probability problems
- Reducing overall score by significant margins
🎯 Mistake Navigation
📊 Fundamental Formulas Quick Reference
Permutations (Order Matters)
$$^nP_r = \frac{n!}{(n-r)!}$$
Arrangements with restrictions need careful case analysis
Combinations (Order Doesn't Matter)
$$^nC_r = \frac{n!}{r!(n-r)!}$$
Selection problems - watch for identical objects
Mistake 1: Confusing Permutations with Combinations
❌ The Wrong Approach
Students use permutations when combinations are needed, and vice versa.
Example: "In how many ways can we select 3 students from 10 for a committee?"
Wrong: $^{10}P_3 = 720$ ❌ (This counts different orders as different)
✅ The Correct Approach
Key Question: Does order matter?
Committee selection: Order doesn't matter (A,B,C same as C,B,A)
Correct: $^{10}C_3 = 120$
When to use Permutations: Arrangements, passwords, ranks
When to use Combinations: Committees, teams, selections
💡 Prevention Strategy
- Ask: "If I change the order, do I get a different arrangement?"
- If YES → Use Permutations
- If NO → Use Combinations
- Remember: Selection = Combinations, Arrangement = Permutations
Mistake 2: Overcounting Repetitions
❌ The Wrong Approach
Students count the same arrangement multiple times due to repetition.
Example: "How many words can be formed from MISSISSIPPI?"
Wrong: $11! = 39,916,800$ ❌ (Doesn't account for identical letters)
✅ The Correct Approach
Account for identical objects:
MISSISSIPPI has: 1 M, 4 I's, 4 S's, 2 P's
Correct: $\frac{11!}{4!4!2!} = 34,650$
General Formula: $\frac{n!}{n_1!n_2!...n_k!}$ for identical objects
💡 Prevention Strategy
- Always check for identical objects/letters
- Use the multinomial coefficient formula
- Divide by factorial of repetitions for each type
- Practice with words like "BANANA", "MATHEMATICS"
Mistake 3: Identical vs Distinct Objects Confusion
❌ The Wrong Approach
Students treat identical objects as distinct, leading to overcounting.
Example: "In how many ways can we distribute 10 identical chocolates among 4 children?"
Wrong: $4^{10}$ ❌ (Treats chocolates as distinct)
✅ The Correct Approach
Stars and Bars method:
For identical objects distributed to distinct recipients:
Number of non-negative solutions to $x_1 + x_2 + ... + x_r = n$
Formula: $\binom{n + r - 1}{r - 1}$
Correct: $\binom{10 + 4 - 1}{4 - 1} = \binom{13}{3} = 286$
💡 Prevention Strategy
- Identify if objects are identical or distinct
- For identical objects → Use Stars and Bars
- For distinct objects → Use $r^n$ or permutations
- Remember: Chocolates, balls = often identical
Mistake 4: Gap Method Errors in Arrangements
❌ The Wrong Approach
Students incorrectly apply the gap method when objects shouldn't be together.
Example: "Arrange 5 boys and 3 girls so that no two girls sit together"
Wrong: $5! \times 4!$ ❌ (Wrong gap calculation)
✅ The Correct Approach
Proper gap method:
Step 1: Arrange the boys: $5! = 120$ ways
Step 2: Create gaps: _ B _ B _ B _ B _ B _
Number of gaps = 5 + 1 = 6 gaps
Step 3: Choose 3 gaps from 6: $\binom{6}{3} = 20$
Step 4: Arrange girls in chosen gaps: $3! = 6$
Correct: $5! \times \binom{6}{3} \times 3! = 120 \times 20 \times 6 = 14,400$
💡 Prevention Strategy
- Always arrange the unrestricted objects first
- Count gaps carefully: n objects create n+1 gaps
- Use combinations to select gaps, then permutations for arrangement
- Practice with different "not together" problems
Mistake 5: Circular Arrangement Confusion
❌ The Wrong Approach
Students treat circular arrangements like linear arrangements.
Example: "In how many ways can 6 people sit around a circular table?"
Wrong: $6! = 720$ ❌ (Counts rotations as different)
✅ The Correct Approach
Circular permutations formula:
For circular arrangements, fix one person's position to account for rotational symmetry
Formula: $(n-1)!$ for distinct objects
Correct: $(6-1)! = 5! = 120$
When clockwise ≠ anticlockwise: $\frac{(n-1)!}{2}$
💡 Prevention Strategy
- For circular tables → Use (n-1)!
- For circular arrangements where clockwise = anticlockwise → Use $\frac{(n-1)!}{2}$
- Fix one position to eliminate rotational symmetry
- Remember: Circular ≠ Linear arrangements
5 More Critical P&C Mistakes
Mistake 6: Inclusion-Exclusion Errors
Forgetting to subtract overlapping cases when using "at least" or "at most" conditions.
Mistake 7: Binomial Theorem Misapplication
Confusing $(a+b)^n$ expansion with general combinatorial problems.
Mistake 8: Probability Connection Errors
Mishandling P&C when transitioning to probability calculations.
Mistake 9: Division into Groups
Forgetting to divide by factorial when groups are identical in size.
Mistake 10: "Beggar's Method" Confusion
Incorrect application of distribution principles for identical objects.
Full detailed solutions for all 10 mistakes available in our premium P&C course
📝 P&C Self-Assessment Checklist
Check which mistakes you're likely to make:
Note: If you checked 3 or more, focus on P&C in your revision plan!
🛡️ Comprehensive P&C Prevention Plan
Before the Exam:
- Practice decision trees for P&C problems
- Create formula cards for special cases
- Memorize the 5-step P&C process:
- Identify if objects are identical/distinct
- Determine if order matters
- Check for restrictions
- Choose correct formula/method
- Verify for overcounting/undercounting
During the Exam:
- Always read carefully - "arrange" vs "select"
- Use case analysis for complex restrictions
- Double-check identical objects
- Verify answer with small numbers when possible
- If stuck, try complementary counting
🎯 Test Your P&C Understanding
Try these problems while consciously avoiding the common mistakes:
1. How many 4-digit numbers can be formed using 1,2,3,4,5,6,7 without repetition?
2. In how many ways can 5 identical prizes be distributed among 8 students?
3. Arrange 4 boys and 3 girls in a row so that no two girls sit together
Master Permutations & Combinations!
These mistakes are common but completely fixable with systematic practice