Standard Selection Problems: The Combinatorics Toolkit
Master committee formations, team selections, and ball picking problems with systematic approaches.
Why Selection Problems Matter in JEE
Selection problems in combinatorics appear in every JEE paper and test your ability to break down complex word problems into simple combinations and permutations. Mastering these will help you:
- Solve 3-6 mark problems in under 5 minutes
- Develop systematic thinking for complex scenarios
- Avoid common pitfalls like double counting
- Build conceptual clarity for probability problems
🎯 Quick Navigation
Essential Combinatorics Formulas
Fundamental Formulas
Combination Formula:
Use when order doesn't matter
Permutation Formula:
Use when order matters
🎯 Decision Framework
- Selecting committee members
- Choosing teams without specific roles
- Picking balls from a bag
- Assigning specific roles (captain, vice-captain)
- Arranging people in a line
- Forming numbers with digits
1. Committee Formation Problems
Problem 1: Simple Committee Selection
From 10 people, in how many ways can we select a committee of 4 people?
Solution Approach:
Step 1: Identify this as a combination problem (order doesn't matter)
Step 2: Apply combination formula:
Step 3: Verify: No restrictions, no specific roles
Answer: 210 ways
Problem 2: Committee with Restrictions
From 7 men and 5 women, select a committee of 5 with at least 3 men.
Solution Approach:
Step 1: Break into cases based on "at least 3 men":
• Case 1: 3 men + 2 women
• Case 2: 4 men + 1 woman
• Case 3: 5 men + 0 women
Step 2: Calculate each case:
Case 1: $C(7,3) × C(5,2) = 35 × 10 = 350$
Case 2: $C(7,4) × C(5,1) = 35 × 5 = 175$
Case 3: $C(7,5) × C(5,0) = 21 × 1 = 21$
Step 3: Add all cases: $350 + 175 + 21 = 546$
Answer: 546 ways
2. Team Formation Problems
Problem 3: Team with Specific Roles
From 12 players, select a team of 5 with specific roles: captain, vice-captain, and 3 players.
Solution Approach:
Step 1: Select captain and vice-captain (order matters):
Step 2: Select remaining 3 players from remaining 10 (order doesn't matter):
Step 3: Multiply: $132 × 120 = 15,840$
Answer: 15,840 ways
Problem 4: Dividing into Multiple Teams
Divide 10 different people into two teams of 5 each. In how many ways?
Solution Approach:
Step 1: Select first team of 5 from 10:
Step 2: The remaining 5 automatically form second team
Step 3: Since teams are unlabeled, divide by 2 to avoid double counting:
Answer: 126 ways
3. Ball Selection Problems
Problem 5: Selecting Identical Balls
A bag contains 8 identical red balls and 6 identical blue balls. In how many ways can we select 5 balls?
Solution Approach:
Step 1: Since balls are identical, we count distributions:
Step 2: Let r = number of red balls, b = number of blue balls
Step 3: r + b = 5, with 0 ≤ r ≤ 8, 0 ≤ b ≤ 6
Step 4: Possible (r, b) pairs: (5,0), (4,1), (3,2), (2,3), (1,4), (0,5)
Step 5: But (0,5) is invalid since we only have 4 blue balls maximum
Step 6: Valid pairs: (5,0), (4,1), (3,2), (2,3), (1,4)
Answer: 5 ways
Problem 6: Selecting Distinct Balls with Conditions
A bag has 5 distinct red balls, 4 distinct blue balls, and 3 distinct green balls. Select 6 balls with at least 2 of each color.
Solution Approach:
Step 1: Start with minimum required: 2 red + 2 blue + 2 green = 6 balls
Step 2: Since we need exactly 6 balls, this is the only possibility
Step 3: Select 2 red from 5: $C(5,2) = 10$
Step 4: Select 2 blue from 4: $C(4,2) = 6$
Step 5: Select 2 green from 3: $C(3,2) = 3$
Step 6: Multiply: $10 × 6 × 3 = 180$
Answer: 180 ways
⚠️ Common Mistakes to Avoid
Conceptual Errors:
- Using permutation when combination is needed
- Forgetting to divide by 2 for identical teams
- Missing "at least" or "at most" conditions
- Double counting identical objects
Calculation Errors:
- Not simplifying factorial expressions
- Missing cases in case-based problems
- Incorrect application of addition/multiplication principles
- Confusing C(n,r) with P(n,r) formulas
🔧 Systematic Problem Solving Framework
Step 1: Identify the Type
- Committee formation → Combination
- Team with roles → Permutation + Combination
- Ball selection → Check if objects are identical or distinct
Step 2: Note Restrictions
- "At least" → Break into cases
- "At most" → Use complement if easier
- Specific inclusions/exclusions → Adjust total count
Step 3: Apply Formulas
- Use combination C(n,r) when order doesn't matter
- Use permutation P(n,r) when order matters
- Multiply for independent choices
- Add for mutually exclusive cases
Step 4: Check for Overcounting
- Divide by k! if k objects are identical
- Check if teams/groups are labeled or unlabeled
- Verify each case is distinct
📝 Practice Problems
Test your understanding with these problems:
1. From 8 men and 7 women, select a committee of 6 with exactly 4 men.
2. In how many ways can 12 different books be distributed equally among 3 students?
3. A box contains 5 red, 4 blue, and 3 green distinct balls. Select 4 balls with at least one of each color.
📋 Quick Reference Guide
Key Formulas:
- $C(n,r) = \frac{n!}{r!(n-r)!}$
- $P(n,r) = \frac{n!}{(n-r)!}$
- $C(n,r) = C(n,n-r)$
- $P(n,r) = r! × C(n,r)$
Common Scenarios:
- Committee without roles → Combination
- Team with captain → Permutation × Combination
- Identical objects → Distribution counting
- "At least" conditions → Case analysis
Master Combinatorics for JEE Success!
These selection problems are guaranteed in every JEE paper. Practice systematically and build confidence.