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Counting Principles Review
1. A magazine poll sampling 100 people gives that following results:
17 read magazine A
18 read magazine B
14 read magazine C
8 read magazines A and B
7 read magazines A and C
9 read magazines B and C
5 read all three magazines
a) How many of the people polled do not read any of the three magazines?
b) How many people read just magazine C?
2. Five cards are dealt from a deck of fifty-two cards.
a) How many different hands can be dealt?
b) How many hands will contain all face cards?
c) How many hands will contain no face cards?
d) How many hands will contain only spades or clubs?
3. A tennis player has 3 pairs of shoes, 6 pairs of socks, 5 pairs of shorts and 8 shirts. How many different
outfits can he wear?
4. a) How many 5 letter “words” can be made using letters from the word LOGARITHM?
b) If the G must be used and it must be in the second position, how many 5 letter words are there?
5. A person walks out of a store having purchased some of the six books she had been looking for. How many
different purchases could she have made?
6. Determine the number of divisors of 3960.
7. On the circumference of a circle 12 points are located and joined in all possible ways.
a) How many chords are formed?
b) How many triangles with chords as sides are formed?
8. How many diagonals does a 14-sided polygon have?
9. Find the number of ways in which 16 different objects can be divided into 3 parcels
containing 4, 5, and 7 objects.
10. Six persons attend a party. How many handshakes will occur if each person shakes hands with every other
person at the beginning and end of the party?
11. How many different 8-letter words can be formed using letters from the word CANADIAN so that D always
immediately precedes an A?
12. How many bridge hands of 13 cards are there which have a 6-card suit, as well as another 5-card suit, as well
as a 2-card suit, which must include the ace of that suit?
13. How many committees are possible from a group of 5 men and 3 women if there are to be four people on the
committee, at least one of whom must be a woman?
14. In how many ways can 6 cars be line up if they are all different and the Chev must be next to the Ford?
15. In how many different ways could you stack 4 quarters, 3 dimes, 2 nickels and one penny?
16. Find the number of 4 letter “words” that can be formed by using the letters of the word FACETIOUS if:
a) if there are no restrictions.
b) if at least one vowel must be included.
c) if exactly one vowel must be used.
17. A club consists of 7 women and 5 men. In how many possible ways can the club select a president, vicepresident, and a secretary if:
a) the president must be a woman and the vice-president must be a man?
b) the president and the vice-president cannot be both men or both women?
18. How many 8 letter words can be formed from the letters in vacation if:
a) there are no restrictions?
b) vct must be together?
c) vowels and consonants must alternate?
19. In how many ways can 7 cars be lined up if either the black Chrysler or the two Cadillac’s (one pink and one
white) must be at the end nearest the road?
20. There are six seats in a car. In how many ways can 6 people be seated in a car if only 3 of them are able to
drive?
21. Given that there are 3 Caramilk bars, 4 Mars bars, and two Peanut Butter Cups,
a) how many different purchases can be made(he must buy something)?
b) how many purchases will contain at least 2 Mars bars?
22. a) How many diagonals does a polygon have if it has 8 sides?
23. A yoga group consists of 8 males and 10 females. In how many ways can a committee of 7 people be formed
from this group if:
a) at least one person must be female?
b) there must be more males than females?
24. The prime factorization of 300 is 2×2×3×5×5. Find:
a) the number of divisors of 300
b) the number of divisors that are composite numbers
c) the number of divisors that are divisible by 5.
25. Write (n + 1)!(n2 + 5n + 6) as a single factorial.
26. In a survey of 56 Mathematics students who wrote the Algebra, Calculus and Finite examinations:
33 passed Algebra
26 passed Calculus
24 passed Finite
10 passed Algebra and Calculus
8 passed Algebra and Finite
14 passed Calculus and Finite
4 passed all three exams
a) Illustrate this information on a Venn diagram.
b) How many students passed Calculus but not Finite?
c) How many students failed all the examinations?
27. From a group of four women and four men, how many different combinations of four can be formed with:
a) no restrictions?
b) four women?
c) three women and one man?
d) two women and two men?
e) four men?
28. In how many ways can 3 adjacent doors be painted if there are 6 different colours of paint available?
29. In how many ways can 5 boys and 5 girls be seated alternately in a a row of 10 chairs, if a boy always occupies
the first chair?
30. The roster of a hockey team contains 10 forwards, five defencemen, and two goalies. How many different
teams can a coach select if he must select three forwards, two defencemen and one goalie?
31. In how many ways can a party of 13 be made up from a group of five adults and 12 children if the party must
contain at least two adults?
32. In a school with 480 girls and 520 boys, how many formal committees of size 5 can be formed that have
more girls than boys?
33. How many committees of three people can be formed from a group of ten people?
34. How many five card poker hands can be made from a deck of 52 cards?
35. A 10-volume encyclopedia sits on a shelf. In how many ways can the 10 volumes be arranged on the shelf so
that some or all of the volumes are out of order?
36. How many different license plate designations can be made using 3 letters followed by a 3-digit number?
37. Seven students pose for a photograph.
a) In how many ways can they be placed in a line?
b) In how many ways can they be placed 4 in the front and 3 behind?
38. A teacher asks a question to a class of 20 students. In how many ways can the class respond ( that is nobody
answers, one per answers etc…)?
39. A committee is to be made up of three girls and three boys. There are five girls and seven boys to choose
from. How many different committees can be formed?
40. There are 16 people at a Christmas party. How many handshakes must be made as they all say goodnight to
each other?
41. A team for a math contest will be chosen according to the following rules:
i)
one student from grade 9 or 10
ii)
one student from grade 11
iii)
three students from grade 12
At Victoria Park this year, there are 3 interest students from grade 9, two in grade 10, four in grade 11 and
three in grade 12. How many different teams could be formed?
42. In how many ways could a jack or a heart be selected from a deck of cards?
43. Draw a tree diagram representing the make-up of a family with three children. How many families have
either a boy as the eldest or a girl as the middle child?
44. There are eight choices for condiments for a hamburger. How many different burgers could be made?
45. I have three pennies, two dimes and four quarters. How many different sums of money could I make? ( zero
is not a sum)
46. Inn which word will you find the greatest numbers of arrangements of all its letters: BINGO, AARVARK,
DEEDED?
47. Simplify:
a) 12 x 11! b)
( n + 3)!
( n − 1)!
c)
12!
12!
+
d) n [n! + ( n – 1 )! ]
8! 4! 9! 3!
48. A club consists of 6 men and 5 women. In how many possible ways can the club select a President, VicePresident and a secretary if:
a) the president must be a women and the vice must be a man?
b) Both the Pres. and vice must be women and the secretary a man?
c) The Pres. and vice cannot both be men or both women?
49. Touchdown!! After a touchdown the 12 players on the field give each other butt slaps. How many slaps were
given?
50. a) How many four letter words can be formed from the letters A, B, C, D, E, if A is always included and no
letter is used twice?
b)How many five letter words can be formed from the letters A, B, C, D, E, if the A and B must be together
and no letter is to be used twice?
51. If the letters of the word SPECIAL are written in every possible way(using all the letters) , how many of
them will not begin with SP?
52. A company disk collection has 6 rock, 8 classical and 7 jazz albums. If a friend asks to sample the collection
by choosing 2 disks of each type how many different samples can she take?
53. It is time to study and there are 10 questions. How many groups of questions can I attempt?
54. Evaluate each of the following, using a formula and showing all steps:
a)
 20   20   20 
 20   20 
  +   +   + …… +   +  
0  1  2 
 19   20 
b)
8  8  9
 + + 
3  4  5 
7  7  7  7  7  7  7  7 
55. Evaluate   −   +   −   +   −   +   −  
0  1  2  3 4 5 6 7 
n 
56. Write in the form   :
r 
 19   19   20 
a)   +   +  
3 4  5 
2
b)
2
2
2
5 5 5
5
  +   +   + .. +  
0
1
2
     
5
57. Five playing cards are face down. Two cards are drawn at random, from the five, how many different hands
can be formed?
58. Four people have been invited to a play. In how many ways can the invitations be accepted?
 n  n 
n 
59.   +   + .. +   = 512 . Find n.
0  1 
n 
60. How many different paths will spell the word binomial in the following diagram:
B
I
N
I
N
O
N
O
M
I
A
I
A
L
A
L
L
61. After pay day my wallet contains 5 twenties, 3 twos and 1 five. How many different sums of money can be
formed from these bills?(zero is not a sum)
62. At a bus stop 10 people get on the bus. If there are only three seats in which left to sit, how many different
groups of three get seated?
63. Find the number of divisors, other than one, of 2700?
64. At a family reunion everyone greets each other with a kiss, on the lips. If there are 20 people at the
reunion, how many kisses take place?
65. In how many different ways can all the letters of the word CHROMATIC be written:
a) without changing the position of any vowel?
b) Without changing the order of the vowels?
66. Two friends decide to play squash twice a week. In how many ways can they select the two days of the week
to play on?
67. Evaluate each of the following, using a formula
a)
b)
9 9 9
9 9
  +   +   + …… +   +   =
 0  1  2 
8  9
 6  6  7 
 + +  =
2  3   4 
68. In the arrangement of letters given, how many paths from top to bottom spell “quotient”?
Q
U
U
O
O
O
T
T
T
T
I
I
I
E
E
N
N
N
T
T
69. On Halloween a trick or treater has a choice of 5 treats from the one house. The little monster can choose
all the treats or just the ones he want. How many choices could the child make if he is sure to take at least
one?
70. A committee of five is to be chosen from 6 students and 7 teachers:
a) no restrictions
b) no teacher is chosen
c) at least one teacher is chosen
71. Canadian postal codes consist of six characters of three letters alternating with three digits. An example
of a postal code is M5N 2R6.
a) How many possible postal codes can be formed if all ten digits and all twenty-six letters are used?
b) How many postal codes can there be if the number 0 and the letters O and I are excluded?
72. The Wimbledon men’s tennis finals consist of at most five sets played between two players. When one player
wins three sets, the match concludes and that player is declared the champion. Use a tree diagram to
illustrate the possible ways in which a match can proceed?
73. There are 4 roads leading from town A to town B and 5 roads leading from town B to town C. In how many
ways can one make a trip from A to C by way of B?
74. There is going to be a volleyball tournament at a beach party. In how many ways can the 20 party-goers be
divided into five member teams?
75. If a + b = -3, evaluate the following:
4 4 4 3
4 2 2 4 3  4 4
  a +   a b +   a b +   ab +   b
0
1 
2 
3 
4
Answers
1)
2)
3)
4)
5)
6)
7)
a) 70 b) 30
 52 
a)  
5 
3x6x5x8
a) P(9,5)
26-1
4x3x2x2
 12 
a)  
2
8)
 14 
  − 14
2
9)
 16   12  7 
    
 4   5  7 
 40 
c)  
 5 
 12 
b)  
5
 26 
d)  
5
b) 8x1x7x6x5
 12 
b)  
3
 6
10)   x 2
2
7!
11)
2!2!
 4  13   3   13   2   1  12 
12)    ×     ×     
 1   6   1   5   1   1  1 
8 35
13)   −    
4 04
14) 5! X 2
10!
15)
4!3!2!
16) a) P(9,4)
17) a) 7x5x10
8!
18) a)
2!
19) 1 x 6! + 1 x 1 x
20) 3x5x4x3x2x1
21) a) 4x5x3-1
8
22)   − 8
2
b) P(9,4) – 4!
b) 7x5x10 + 5x7x10
6!× 3!
b)
2!
5! x 2!
c)
4!× 4!× 2
2!
b) 4x3x3
 18   8   10 
23) a)   −    
b)
 7  7  0 
24) a) 3x2x3
b) 18-1-3
25) (n+3)!
26) b) 12 c) 1
4
8
27) a)  
b)  
4
4
28) 6x6x6
29) 5! x 5!
 10  5  2 
30)    
 3  2  1 
c) 5x4x3x2x4
 8  10   8   10   8  10   8  10 
   +     +    +   
 4  3   5   2   6  1   7  0 
c) 3x2x2
 4  4 
c)   
 3  1 
 4  4 
d)   
 2  2 
4
e)  
4
 17 
 5  12 
31)   − 0 −   
 13 
 1  12 
 480   520   480   520   480   520 
32) 



+
+

 3  2   4  1   5  0 
 10 
33)  
3
 52 
34)  
5 
35) 10! – 1
36) 26x26x26x10x10x10
7 
37) a) 7!
b)   × 4!× 3!
4
20
38) 2
57 
39)    
3 3
 16 
40)  
2
41) 5x4x1
42) 4 + 13 – 1
43) 6
44) 28
45) 4 x 3 x 5 – 1
 6! 
 8! 
46) Bingo (5!) Deeded 
Aardvark 


 3!3! 
 3!2! 
47) a) 12!
b) (n+3)(n+2)(n+1)(n)
c) 715
d) (n+1)!
48) a) 5x6x9
b) 5x4x6
c) 11x10x9 – 6x5x9 - 5x4x9
 12 
49)   × 2
2
50) a) 4x4x3x2
b) 4! X 2
51) 7! – 5!
 6  8   7 
52)     
 2  2   2 
53) 210
 10 
54) a) 220
b)  
5
55) 0
 21 
 10 
56) a)  
b)  
5
5
5
57)  
2
58) 24
59) 9
60) 18+18+6
61) 6 x 4 x 2 – 1
 10 
62)  
3
63) 3x4x3-1
 20 
64)  
2 
6!
65) a)
2!
7 
66)  
2
b)
67) a) 29
8
b)  
4
9!
3!2!
68) 30+30
69) 25-1
 13 
 6  7 
70) a)  
b)    
5
5 0 
71) a) 26x10x26x10x26x10
72)
73) 4x5
 20   15   10  5 
74)       
 5   5   5  5 
75) ( −3) = 81
4
 13   6  7 
c)   −   
 5   5  0 
b) 24x9x24x9x24x9
ANSWERS
1. a) 70 b) 3
2. a) 2598960 b) 792
c) 658008 d) 65780
3. 720
4. a) 15120 b) 1680
5. 63
6. 48
7. a) 66 b) 220
8. 77
9. 1441440
10. 30
11. 1260
12. 636045696
13. 65
14. 240
15. 12600
16. a) 3024 b) 3000 c)
480
17. a) 350
b) 700
18. a) 20160 b) 2160 c)
576
19. 960
20. 360
21. a) 59
b) 36
22. a) 20
b) 11
23. a) 31816 b) 11208
24. a) 18
b) 14
c) 12
50.
25. (n + 3)!
51.
26. b) 12
c) 1
52.
27. a) 70 b) 1 c) 16 d)
53.
36 e) 1
54.
28. 216
55.
29. 14400
30. 2400
56.
31. 2375
 480   520   480   520   480

57.
32. 



+
+
 3   2   4   1   558.

59.
33. 120
60.
34. 2598960
61.
35. 3628799
62.
36. 17576000
63.
37. a) 5040 b) 5040
64.
38. 1048576
65.
39. 350
66.
40. 120
67.
41. 20
68.
42. 16
69.
43. 6
70.
44. 256
71.
45. 59
46. aardvark
72.
47. a)12! b)
73.
[(n+3)(n+2)(n+1)n] c)
74.
75.
715 d) (n+1)!
48. a) 270 b) 120 c) 540
49. 132
a) 96 b) 48
4920
8820
1024
a) 1048576 b) 252
0
 21 
 10 
b)  
a)  
5
5
10
16
n=9
42
47
120
35
190
a)360 b) 30240
21
512,70
60
31
a) 1287 b) 6 c) 1281
a) 17576000
b) 10077696
diagram
20
1.17 x 1010
81
`