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
7.
Permutations of Non-
Distinguishable Objects
The number of ways to arrange n objects that contain
sets of non-distinguishable objects is given by:
8.
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 ∴
9.
How many different quot;wordsquot; can you make from the letters
in the word STATISTICS?
10.
How many distinguishable ways can 3 people be seated
around a circular table?
11.
How many distinguishable ways can 4 people be seated
around a circular table?
12.
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
13.
How many distinguishable ways can 3 beads be
arranged on a circular bracelet?
14.
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
15.
How many distinguishable ways can 4 beads be
arranged on a circular bracelet?
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