Circular permutations and applications.

1. (a) How many “words” of 4 different letters each can be made from the
letters A, E, I, O, R, S, T?
(b) How many of these words begin with a vowel and end with a
consonant?
(c) In how many of these words do vowels and consonants alternate?

2. How many distinguishable ways can 3 people be seated around a
circular table?

3. Circular Permutations
The number of ordered arrangements that can be made of n objects in a
circle is given by:
(n - 1)!
Example: How many different ways can 6 people be seated around a
circular table?
(6 - 1)! = 5!
= 120
Special Case: A bracelet is a circle that can be flipped over. The number of
different arrangements that can be made of objects on a bracelet is:
Example: How many bracelets can can be made from 6 different beads?
(6 - 1)! = 5!
2 2
= 60

4. In how many ways can 4 married couples seat themselves around a circular
table if:
(a) spouses sit opposite each other?
(b) men and women alternate?

5. KAS
Emilia

6. Mom and dad and their 6 children (3 boys and 3 girls) are to be seated at a
circular table. How many ways can this be done if mom and dad sit together
and the males and females alternate?

7. (a) How many numbers of 5 different digits each can be formed from
the digits 0, 1, 2, 3, 4, 5, 6?
(b) How many of these numbers are even?
(c) How many of these numbers are divisable by 5?

8. (a) How many numbers of 5 different digits each can be formed from
the digits 0, 1, 2, 3, 4, 5, 6?
(b) How many of these numbers are even?
(c) How many of these numbers are divisable by 5?

9. (a) How many numbers of 5 different digits each can be formed from
the digits 0, 1, 2, 3, 4, 5, 6?
(b) How many of these numbers are even?
(c) How many of these numbers are divisable by 5?

10. How many different necklaces can be made from 12 beads of different
colours?

11. How many different necklaces of 12 beads each can be made from 18 beads
of different colours?

12. There are 9 chairs in a row. In how many ways can 4 students be seated in
consecutive chairs?
(Hint: First find the number of ways of choosing 4 consecutive chairs.)

13. In how many ways can 8 books be arranged on a shelf, if 3 particular books
must be together?