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Date:
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MDM 4U UNIT 1 (CHAPTER 4 TEST)
PERMUTATIONS and ORGANIZED COUNTING
K&U
APP
COM
TIPS
22
19
10
14
Factorial: n! = n x (n - 1) x (n -
Key Equations:
2)x... x3 x2 X 1
n!
Permutations: n Pr =
(n r)!
-
Permutations with Identical Items.
1. Write as a single factorial. (4 K)
a. 3X 2 X 4 X 5 X7 X 6
n!
a!b!c!...
3. Solve for n. (4 K, 2 A)
2(nP5) =
-1I
•
2L
(n-5)1
n-1P6
(1")
GI- -
=
(n-
41
2. [r r‘1
b. 72 X 7!
(r‘ - 1)
= 9 )(SX -1 !
Z
/
(n (.1.116(A-A(t -776(n-At-ep
-
c.
)Cyl-Acrybo-A111-061-()64.
(n 2-9n+20)(n-6)!
(
n - min
-5)0-0 1.
2n
(n-k1)‘
2. Evaluate. (3 K)
15!
a.
3!8!
01-s-')(o-6)
2n=n 2 -IIn ~ 3
n 1*- 13n + 3o
3)
o = (n
(
.5 3
ismyx13x‘2.?“1)4.€5
.3')< -ZX
/
4. Simplify. (2 K)
(n — 1)!
(n2 — n)(n — 2)!
ks-AkLi x13X12v-IIK
5 44051-40
b.
•
7P3(25 -4P2) + 7!/3! + 11(7P3 + 0!)
Z1° (
zt o
%.2-) 4 (840) -1.- 11
+ gip
(2/0-1)
z 3 2... 1
5S9 I
If)
o
n=Z
n=
Name:
5.
Date:
How many different ways can you
8. Kenya has a die and a coin. She rolls the die
arrange all of the following items in a
and then tosses the coin. Make a tree
line? (1K)
diagram to show the possible outcomes.
How many different outcomes are there?
(2K,1A)
4104- t©44x:x0)-->- ©40
4.9 (31.2.
1 81
toz,12,12)
6.
_ G 2210Zo8oO
2-4-1S2 4
19 2
VDSSA:11 ‹.
0 0
A Chinese restaurant features a lunch
special with a choice of wonton soup or
spring roll to start, sweet and sour
chicken balls, pork, or beef for the main
dish, and steamed or fried rice as a side
dish. Create a tree diagram to show all
the possible lunch special at this
restaurant. How many different
possibilities are there? (2K,1A)
5 SCA3
f›rt
SeF.(2-
4.
ol(-Gierri
foss% b 141es
Z
9. In how many ways can you roll a sum of 6 or
a sum of 10 with a pair of dice?(1A)
4-1 s- (4,
I 3 LI s7
3 LI s2. 3
SkII
2
3
5r •
-
8..
watts .
s
fe)
rb
5- 1
-7
TVN-ert_ a_ce
2 (ID
8 9 S
2,
10. In how many ways can a set of eight books
7. Canadian postal codes consist of
alternating 3 letters and 3 numbers (L#L
#L#). How many different postal codes
be arranged on a shelf so that volumes one
and two are beside one another? (2A, 1C,
1T)
vt we cor V•0..vc_ k Alen 2. or 2.-Pra.n
X 2 ----
are possible? (2A)
LI ea Y.2.
= 1 06 $3, b
2- (0 X 1b)t. 2/0•4-10t240
L
1
S- 1
(o
(:) ( 0
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11. In how many ways could you arrange a
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14. Bill works in an ice cream store for the
display of stationary supplies consisting
summer months. How many different cones
of 14 notebooks, 5 packages of lined
can Bill create if he has chocolate, mint
paper, and 50 pens if all the items are laid
chocolate chip, vanilla, maple walnut, and
pistachio ice cream available and a cone can
out in a row? (1A)
have at most 3 scoops?
n = 69
--ro+0,A
G91
ctaAJC,u-r- S = 5
-to
Is- p/
ryl
b -C-
(4-r,1c)
:7 5 .5g ( X tC)
3
scoops = 5P 3 = (00
Swo9S = 51° 2_ = 2_0
be3olnir43-:
-ey
0
53
z'2 1
■
4-f
.; 5
scboV
12. How many permutations of the word
committee begin or end with an e? (2T)
c- ov-kereak- cones = (3o-1- zuts
_ 10 08 0
$S
!
-1-4(vCre_ 0--re
lobgbi ioc
c/.1:-Feete-M- Clyne_s‘
201(00
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-1.-Vvere_ ace_ zo% (13 o etrOlu-i-cklI6115
13. Seven children are to line up for a
15. How many possible arrangements are there
photograph.
a.
for the letters in the work BANANA? (1A)
How many different arrangements are
possible?(1K)
4, I
z)-6,
!
= Go
16. Express as a single term from Pascal's
b.
How many arrangements are possible if
Brenda is in the middle?(1A)
triangle. (2K)
a.
ti2,4 + t12,5 = -€.
b.
t14,4 t13,4 =
z
•
%3
Name:
Date:
17. Casey has 5 trophies. In how many
20. A checker is placed on a checkerboard as
different arrangements can she put the
shown. The checker may move diagonally
trophies on a shelf if she must put on at
upward on the white squares only. It cannot
least three trophies on the shelf? (3T,1C)
rekort- tro(Aies
3 of
s P3 = (oo
move into a square with an X, however, it can
jump over the X into the diagonally opposite
square. How many paths are there to the top
row of the board? (2A)
5Py =17_O
S p y -= 120
61::> fi 12O r 120 = 3 4:) Ll di-QC-emit
wai6s
18. Which row of Pascal's triangle has terms
that sum to 524288? (1K)
)(
z-- SV-128g
X t o3 2 = lus 5 2 y egg
=I
12i- i2+11 = 4S Fo -t-V■ S.
19. A university has a telephone system in
which extension numbers are three digits
long with no repeated digits and no Os.
a.
21. Calvin is trying to get home without running
The university has 492 telephones at
into Moe the bully (who will steal his lunch
present and is planning to add another 35
money). However, Calvin does not know what
in the near future. (3A,3C)
path Moe is waiting at. Based on the diagram
Should the university change its system?
Why or why not?
below, how many possible routes are there so
that Calvin won't lose his lunch money
=
Lfctz.+ Sc= 579- ncesAed
e
assuming that he always moves towards his
soq cerk.on
■
► C‘R--C ,
house? (2A)
3
The u_niveirsktt- shay.0.613x.cy
stern bei20-use --1-k40-4C0-11
i+ S
`
,
on l9 Ivvate SW`( diffeXents bus
b.
-ev‘ey
The Drama Department uses extensions
that begin with 3. How many extensions
can the Drama Department have with the
3 + 31 3 3 -- woe s
current system?
(3)
xsKi
-7- 5(0 e_x-i-t ► r -ton
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Date:
Name:
22. Lisa is planning the seating for the head table at a gala. The eight speakers will all be seated along
one side of the table. Richard wants to sit beside Hang, and Lisa knows that Thomas and Lily
should not be seated together, as they have just broken up. In how many ways can Lisa make up
the seating plan? Explain your reasoning. (4C, 4T)
Richard A 1-to..Y\5 act as I kn its 4exe_ are_ now 1 va ►=4-s
(now._ caDo-ra
)( 2_ = 5oLko x2. = too80
cAtka R. (chard, Car be Silt
0
use
,n _vvvc \ 64 or riNit
m ci-h 8.
Tho rras
wc, ►* So A- neck
ceCe_ 110W
6 X 2 X 2- =
IC
ec■fetra, 71/4
fko
1/4_02
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col V-Je._ 61, \ eh-
s
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\(1 -1-
se' 11 \° "1- 5
100 8 0 — 2g 8o -; 7 2-0
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