Combinations④ Conditional Selection・Grouping
Example① (Including or Excluding Specific People)
Choosing 5 people from 10, including both A and B: choose 3 from the remaining 8 → 8C3 = 56 ways
Choosing 5 people from 10, including both A and B: choose 3 from the remaining 8 → 8C3 = 56 ways
Example② (Grouping)
Dividing 9 people into 3 groups of 3 (groups indistinguishable): 9C3×6C3×3C3÷3! = 84×20÷6 = 280 ways
Dividing 9 people into 3 groups of 3 (groups indistinguishable): 9C3×6C3×3C3÷3! = 84×20÷6 = 280 ways
Key Points
- Satisfy the conditions first, then count the remaining choices
- Indistinguishable groups → divide by the factorial of the number of identical groups
- “At least” → use the complementary event
Practice Problems
- How many ways can 4 people be chosen from 5 men and 4 women, including at least 1 person of each gender?
- How many ways can 10 people be divided into two groups of 6 and 4 (groups distinguished)?
- How many ways can 8 people be divided into 3 teams of 4, 2, and 2 (only the 4-person team distinguished)?
Answers
- Total 9C4=126 − men only 5C4=5 − women only 4C4=1 = 120 ways
- 10C6=210 ways
- 8C4×4C2×2C2÷2!=70×6÷2=210 ways