đź‘Ą Committees & Conditions

Problems involving selecting people for roles with various constraints and requirements.

🎯 Recognition Cues

  • Keywords: “choose”, “select”, “committee”, “team”, “president”, “vice-president”
  • Setup: Selecting people from a group for specific roles
  • Constraints: Gender requirements, role assignments, leadership positions

đź“‹ Solution Template

  1. Identify the selection type:

    • Ordered selection: Roles matter (president ≠ vice-president)
    • Unordered selection: Roles don’t matter (committee members)
    • Mixed: Some roles matter, others don’t

  2. Apply the appropriate method:

    • Ordered: Use permutations $P(n,k)$
    • Unordered: Use combinations $\binom{n}{k}$
    • Mixed: Use combinations for unordered parts, permutations for ordered parts

  3. Apply constraints:

    • Gender requirements: Count separately for each gender
    • Role restrictions: Fix specific people in specific roles
    • At least/at most: Use complement counting or casework

đź’ˇ Micro-Examples

Basic Committee

  • Problem: How many ways can you choose a 3-person committee from 10 people?
  • Solution: $\binom{10}{3} = 120$ ways (unordered selection)

Ordered Selection

  • Problem: How many ways can you choose a president and vice-president from 10 people?
  • Solution: $P(10,2) = 10 \cdot 9 = 90$ ways (ordered selection)

Gender Requirements

  • Problem: How many ways can you choose a 3-person committee with at least one woman from 6 men and 4 women?
  • Solution: Total - All men = $\binom{10}{3} - \binom{6}{3} = 120 - 20 = 100$ ways

Mixed Selection

  • Problem: How many ways can you choose a president, vice-president, and 2 committee members from 10 people?
  • Solution: Choose president: 10 ways, choose vice-president: 9 ways, choose 2 members from remaining 8: $\binom{8}{2} = 28$ ways. Total: $10 \cdot 9 \cdot 28 = 2520$ ways

⚠️ Common Pitfalls & Variants

Pitfall: Confusing ordered and unordered selection

  • Wrong: “Choose president and vice-president” = $\binom{10}{2} = 45$ ways
  • Right: $P(10,2) = 90$ ways (order matters: president ≠ vice-president)

Pitfall: Forgetting to account for remaining people

  • Wrong: “Choose president, vice-president, and 2 members” = $P(10,4) = 5040$ ways
  • Right: $P(10,2) \cdot \binom{8}{2} = 90 \cdot 28 = 2520$ ways (president and vice-president are distinct roles)

Pitfall: Misapplying gender constraints

  • Wrong: “At least one woman” = “Exactly one woman” + “Exactly two women” + “Exactly three women”
  • Right: Total - All men = $\binom{10}{3} - \binom{6}{3} = 100$ ways (complement counting)

🏆 AMC-Style Worked Example

Problem: How many ways can you choose a 4-person committee with exactly 2 men and 2 women from a group of 6 men and 5 women?

Solution:

  • Step 1: Choose 2 men from 6: $\binom{6}{2} = 15$ ways
  • Step 2: Choose 2 women from 5: $\binom{5}{2} = 10$ ways
  • Step 3: Multiply: $15 \cdot 10 = 150$ ways

Key insight: When you have multiple independent choices, multiply the counts!

đź”— Related Topics

Next: Balls in Bins → At Least/At Most