Permutations of Non-Distinguishable Objects and Circular Permutations.

1. Permutations of Non-
Distinguishable Objects
and
Circular Permutations
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2. How many four-digit even numbers are there if the same digit cannot
be used twice?

3. How many four-digit even numbers are there if the same digit can be
repeated?

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

5. How many different 4 letter quot;wordsquot; can you make from the
letters in the word BOOK?

6. K O OB
B O OK
BOOK KOOB
BO K O K O BO
BO K O K O BO

8. Permutations of Non-
Distinguishable Objects
The number of ways to arrange n objects that contain
sets of non-distinguishable objects is given by:

9. Example: How many different quot;wordsquot; can be made form the
letters in the word:
(a) BOOK (b) MISSISSIPPI
# of O's = 2 ∴
# of I's = 4
# of S's = 4
# of P's = 2 ∴

10. How many different quot;wordsquot; can you make from the letters
in the word STATISTICS?

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

12. How many distinguishable ways can 4 people be seated
around a circular table?

13. 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

14. How many distinguishable ways can 3 beads be
arranged on a circular bracelet?

15. Circular Permutations
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

16. How many distinguishable ways can 4 beads be
arranged on a circular bracelet?