Circular Permutations
Ways to arrange n objects in a circle → (n−1)! ways (fix one object and arrange the rest)
Necklace permutations (reflections considered identical) → (n−1)!/2 ways
Example ①
Six people in a circle: (6−1)! = 5! = 120 ways
Six people in a circle: (6−1)! = 5! = 120 ways
Permutations with Repetition and Repeated Objects
Arrange r objects selected with repetition allowed from n types → n^r ways
If among n objects, identical objects occur p,q,r… times → n!/(p!×q!×r!×…)
Example ②
Ways to arrange the 6 letters a,a,b,b,b,c: 6!/(2!×3!×1!) = 60 ways
Ways to arrange the 6 letters a,a,b,b,b,c: 6!/(2!×3!×1!) = 60 ways
Key Point
- Circular permutation → fix one person → (n−1)!
- With repetition allowed → n^r
- Repeated objects → divide n! by the product of the factorials of the repetitions
Practice Problems
- How many circular permutations are there for 5 people?
- In a circular permutation of 3 men and 3 women, how many arrangements alternate men and women?
- How many ways are there to arrange the 7 letters in SUCCESS (S:3,C:2,U:1,E:1)?
Answer
- 4! = 24 ways
- Circular permutations of the men 2! × arrangements of the women 3! = 2×6 = 12 ways
- 7!/(3!×2!) = 420 ways