Summary of the Number of Cases
- Multiplication principle, addition principle, and inclusion-exclusion principle
- Permutations nPr = n!/(n−r)!
- Circular permutations (n−1)!, necklaces (n−1)!/2
- Permutations with repetition n^r
- Permutations with identical objects n!/(p!q!…)
Comprehensive Practice Problems
- Arrange 4 men and 3 women. How many arrangements have no two women adjacent?
- How many six-digit integers can be formed using the six digits 0,1,1,2,2,2, with the first digit not 0?
- How many ways can 8 people be divided into 3 teams of 4, 2, and 2 (with the teams distinguished)?
Answers
- Arrange the 4 men (4!=24), then place the women in 3 of the 5 spaces: 5P3=60 → 24×60=1440 arrangements
- Total: 6!/(1!×2!×3!)=60 arrangements. When the first digit is 0: 5!/(2!×3!)=10 arrangements → 50 integers
- 8C4×4C2×2C2=70×6×1=420 arrangements