Name: ___________________________
May 29, 2014
Math 3/4
Ms. Scala/Ms. Kear
Permutations
The idea of arranging objects is an important one in mathematics. We call such
arrangements permutations.
Definition: a permutation is a specific ordering of elements of a set. Permutations
can use all of the elements of a set or just some of them.
Exercise # 1 List all possible permutations of the letters given in the following sets.
Count the total number of permutations for each.
a  B,A
b  B,A,T
c  B,A,T,H
Exercise # 2 Use the Fundamental Counting Principle to help you answer this
question? Katie has chosen 10 songs for her playlist and claims that there are
millions of different ways for her to arrange the songs. Is she correct?
o
When we can to count the number of permutations of all the elements in a set we can
use factorial notation. An example of factorial notation is the following:
10!  10  9  8  7  6  5  4  3  2  1 = 3,628,800
Exercise # 3 Write out and evaluate each of the following:
a  2!
b  3!
 c  4!
 d  5!
Sometimes we want to arrange just a subset of the elements in a set instead of the
whole set.
Permutation Notation: The number of ways to permute r-elements (specific
P
number) from a set of n-elements (total number) is symbolized by n r .
Exercise # 5 Ten people enter a race in which there can be no ties. Different awards
are given for first through third place. In how many different ways can these awards
be given out?
Exercise # 6 A Coach is trying to pick 5 players for distinctly different positions on
a basketball team. He has 9 players that he can pick from. How many different ways
can the coach fill these positions?
Permutations with Repetition
When there are matching elements in the set, we need to compensate for them in
order to get an accurate number of arrangements:
Exercise # 1 Consider the letters in the word BEEPER.
How many total ways are there of rearranging the letters in the word BEEPER if the
3-E’s are considered to be the same?
=
= 120 distinct arrangements
Exercise # 2 Consider the word ASPARAGUS.
How many total distinct arrangements of the letters in this word can be made?
Class Work
1. Find the total number of ways of rearranging the letters in the following words.
Express your answer using factorial notation and also as a whole number.
(a) DADDY
(b) SOCCER
(c) TENNESSEE
(d) CALCULATE
2. The Principal would like to assign each student with a 7-digit code made from the
seven digits in the school’s phone number, 478-4780. If there are 3,200 students in
the high school, will the Principal have enough codes? Numerically justify your
answer.
3. How many 10-digit codes can be made if the Principal also included the area code of
741? Justify your answer with numerical evidence.
Class Work:
1. How many ways are there of rearranging all of the letters in the word MICE?
2. Evaluate each of the following:
a
P
7 4
b 
P
8 3
c
9 2
P
 d
10 1
P
d
10 4
P
3. Which of the following is equivalent to 10  9  8  7?
 a  10!
b 
P
10 3
c
10 7
P
4. How many different 3-arrangements words can be formed from the letters in the
word NUMERAL if there is no repetition allowed?
5. A company would like to create a new password for their Internet service. The
password will consist of 5 different characters from the union of the set of 26
letters A, B, C, ..., Z and the set of 10 digits 0, 1, 2, ..., 9 .
 a  How many different passwords could be created?
b  If this company currently has 25 million Internet customers worldwide, are there
enough passwords?
For the following, find the total number of distinct arrangements of the letters in
the following words. Express your answer in both factorial notation and as a whole
number.
4. CALCULUS
5. PAPA
6. NONSENSE
7. PEPPER
8. SCISSORS
9. PARALLEL
10. MISSISSIPPI
11. SYSTEM
12. BOOKKEER
Name ___________________________
May 29, 2014
Mrs. Scala/Ms. Kear
Math ¾
Permutations HW
1. How many different 6-letter arrangements can be formed using the letters in the
word “ABSENT,” if each letter is used only once?
2. A locker combination system uses three digits from 0 to 9. How many different
three-digit combinations with no digit repeated are possible?
3. How many different five-digit numbers can be formed from the digits 1, 2, 3, 4,
and 5 if each digit is used only once?
4. Which expression represents the number of different 8-letter arrangements that
can be made from the letters of the word "SAVANNAH" if each letter is used only
once?
5. What is the total number of different seven-letter arrangements that can be
formed using the letters in the word "MILLION"?
6. How many different two-letter arrangements can be formed using the letters in
the word "BROWN"?
7 . How many different 11-letter arrangements can be formed from the word
“MISSISSIPPI” if each letter is used only once?