C-CLI
NAME: DATE:
key INTEGRATED ALGEBRA 1
MR. THOMPSON
LESSON 15.9
PERMUTATIONS
HOMEWORK ASSIGNMENT # 118: PAGES 634-635: # 2 - 34 EVENS
EXAMPLE I SEMENEEZZEZEMitracar:.,
Compute the value of each expression.
b. 2P2
c. 7I3!
a. 6!
Solution a. 6!= 6 x 5 x4x 3 x2X 1 = 720
b. 2P2 = 2! = 2 X 1 = 2
ceR 7x6x 3x X 1x X iX
2X1_7X6x5X4 x1x X iXxl_ 840
1
Calculator a. ENTER: 6 VHSWEnriR C. ENTER: 72112
Solution
3 !qv,
W
DISPLAY:
12 tal
840
Answers a.120 b.2 c. 840
EXAMPLE 2
Paul wishes to call Virginia, but he has forgotten her unlisted telephone number. He knows that the exchange (the first three digits) is 555, and that the last
four digits are 1, 4, 7, and 9, but he cannot remember their order. What is the
maximum number of telephone calls that Paul may have to make in order to
dial the correct number?
Solution The telephone nunaber is 555- . Since the last four digits will be an arrangement
of 1, 4, 7, and 9, this is a permutation of four numbers, taken four at a time.
4P4 = 4! = 4 x 3 x 2 x 1 = 24 possible orders
Answer The maximum number of calls that Paul may have to make is 24.
9 n1
EXAMPLE 3
Evaluate 6P2.
Solution This is a permutation of six objects, taken two at a time.
There are two possible formulas to use.
6
P = 6 x 5 = 30
2
Calculator ENTER: 6 tiiiflO
cHci3
6 p2
76d =
xX )
)(
‹
3x
x _ 6 x 5 = 30
2
Solution
DISPLAY:
Answer 6P2 = 30
El
EXAMPLE 4
How many three-letter "words" (arrangements of letters) can be formed from
the letters L, 0, G, 1, C if each letter is used only once in a word?
Solution Forming three-letter arrangements from a set of five letters is a permutation
of five, taken three at a time. Thus:
5 P3 =
5 p3
Answer
= (5 51 3)1
5 X 4 X 3 = 60
or
2,_5x4,2<)342xl_ 5 X 4 X 3 = 60
60 words
EXAMPLE 5 filroa
sptioaraWMI:
, r,
A lottery ticket contains a four-digit number. How many possible four-digit
numbers are there when:
a.a digit may appear only once in the number?
b.a digit may appear more than once in the number?
Solution a. If a digit appears only once in a four-digit number, this is a permutation of
10 digits, taken four at a time. Thus:
1aiD4 = 10 x 9 x 8 x 7 = 5,040
b. If a digit may appear more than once, we can choose any of 10 digits for the
first position, then any of 10 digits for the second position, and so forth. By
the counting principle:
10 x 10 x 10 x 10 = 10,000
Answers
a. 5,040 b. 10,000
Writing About Mathematics
s,
c /Show that n! = n(n — 1)!.
n
r_cr• —1 ) 6
2.. 1-1 c.NS.
'
4) 'Cm -
.k\-1); So
Developing Skills
In 3-17, compute the value of each expression.
-1 • to • C • , S • 2. • t
Li. x3 >c2- X
cosi
6P3
(.
r\
5 , t4.2.
Sn.s.s 4. 3,2.1
d
ol 32.0 \
v., 9991
" (999 - 5))
71)6
11 20132
J. G. S.
ac.
5 •LI
9. BPB
7. (3 + 2)!
5. 7!
3.4!
•3- 2-
9
9S,o Q q, i is7-7; 8 90
120
In 18-21, in each case, how many three-letter arrangements can be formed if a letter is used only
once?
19. TIGER
2L LEOPARD
'PS
193
•
6.0
5'
2\0
Applying Skills
23. Using the letters E, M, I, T: a. How many arrangements of four letters can be found if each
letter is used only once in the "word"? b. List these "words."
(ti
az.,
14 c-tr
i
Estanti frt.1 t artsit Vn.pat1,..-rz4, wit;
v•tsn • •tt• ttA i t-te, TA." n -444n-e
a rrt-1, T-F--tAt 1-0.01
i
i
1
"T'S‘A/c. "Tes.Q./vn srl-tAt l-t-t Et, AttAlte. o
l
\
25. In a game of cards, Gary held exactly one club, one diamond, one heart, and one spade. In
how many different ways can Gary arrange these four cards in his hand?
(--t
.1
•
27. There are 30 students in a class. Every day the teacher calls on different students to write
homework problems on the board, with each problem done by only one student. In how
many ways can the teacher call students to the board if the homework consists of:
a. only 1 problem? b. 2 problems?
c. 3 problems?
245
t
(-3
n
3 0 2.•
I
ea
ao. -2.c‘ . La
elt
30 • Vrk
csio\
d's
2- LAI
Ea C.)\
29. How many different ways are there to label the three vertices of a scalene triangle, using no
letter more than once, when:
a. we use the letters R, S,
S‘k
b. we use all the letters of the English alphabet?
10, CD3
c1S,
6c)o-\
2.G.2S-7-1/44
31.
How many possible ways are there to write two initials, using the letters of the English
alphabe t , if:
a. an initial may appear only once in each pair?
2..co
zs
C:.50,\
b. the same initial may be used twice?
z
Co
G7
G
In 32-35: a. Write each answer in factorial form.
scientific notation.
b.
Write each answer, after using a calculator, in
13. We learn the alphabet in a certain order, starting with A, B, C, and ending with Z. How
cd many possible orders are there for listin the letters of the English alphabet?
?
7.4
crt
?_Cm!
n••n•••••
o 12A 194 ‘1
35.
n1
10 ac
Forty people attend a party at which eight door prizes are to be awarded. In how many
a is ers be announced?
orders can the names I
Co-)
(1._
eiz )
324
3. t oolS 6
Vi ol x,
It
cktk
__)