Math 30P
1.
Permutations and Combinations
True or false?
a) 4(3!) = 4! b) 4 = 5 c) a! = 2!
e) (4!32!, = 8 f) 9(8!)(7!)...(l!) = 9!
d) (3!)(6!) = 9!
A set of 4 books can be arranged on a shelf in only 4
different ways.
--h)--Itis-possible--to form-5!-permutations of-all the letters of
r -LEARN:
--- the wo
i) From a group of 6 people it is possible to form 240
different lists of candidates for the 4 offices: president,
vice-president, secretary, and treasurer.
12. In how many ways can a class of 20 students be divided into
a) 2 groups of 10 students each?
*
b) 4 groups of 5 students each?
13. How many triangles are determined by the vertices of a
regular decagon?
g)
14. A legislative body of 13 people passed a law by a vote of 7
to 6. In how manydifferent ways could their vote have
resulted?
15. From a standard deck of 52 cards, how many different 5-card
hands can be formed if
a) there are no restrictions?
b) each hand has at most 2 red cards?
c) each hand has at least 4 black cards?
d) each hand is either all red or all black?
2.
How many different 4-digit natural numbers can be formed
using the digits 2, 4, 6, and 8 if a digit may be used more
than once in a number?
3.
How many different 4-digit natural numbers can be formed
from the digits 1, 3, 5, 7, and 9 if no digit is repeated in a
er?__.
n
16. A students must choose 10 of 15. problems to complete on a
final exam. In how many ways can the student do this?
4.
How many different 4-letter arrangements are there of the
letters of the word TRADE?
5.
The future Soholt Field at GCHS will eventually be a
stadium with 9 gates: 5 on the east side and 4 on the west
side.
a) In how many ways will a person be able to enter by an
N
east gate and leave by a west gate?
b) -Inhow many ways will a person be able to enter and
leave the stadium?
17. In how many ways can 6 students be seated in a row if
a) there are no restrictions?
b) Kristen and Kaitlyn must sit together?
c) Bill and Nathan must not sit together?
6.
7.
8.
How many different automobile license plates can be made
using 3 letters followed by 3 digits if
a) characters can be repeated?
b) characters cannot be repeated?
How many different signals can be made from 6 differently
colored flags if each signal is to consist of 4 flags hung in a
vertical column and order of the flags does matter?
How many arrangements of 5 different geometry books and 3
different chemistry books can be made if all of the books are
placed on a shelf in such a way that the geometry books are
kept together and to the left of the chemistry books?
9.
How many natural numbers of one or more digits can be
formed using the digits 6, 7, 8, and 9 if a digit may be used
more than once in any number and the number must be
between 5 and 10 000?
10. In how many ways can a game of doubles in tennis be
arranged if each team is to be composed of a boy and a girl
selected from a group of 3 boys and 4 girls?
How many committees of 5 people can be formed from a
group of 12 people?
18. A railway has 50 stations. If the names of the point of
departure and the destination are. printed on each ticket, how
many different kinds of 1-way tickets can be printed?
19. In how many ways can 15 different gifts be divided among
A, B, and C, if A is to receive 2 gifts, B is to receive 3, and C
is to receive 10?
20. A town council is made up of a mayor and 6 aldermen. How
many different committees of 4 can be formed if
a) the mayor is on each committee?
b) the mayor is on no committee?
21. In how many ways can the letters of COMMITTEE be
arranged?
22 Four letters of the word ZEPHYR are to be used to form a
"word". How many of these "words" will
a) not contain the letter H?
b) contain the letter R?
c)
begin with Z and end with R?
23 In how many ways can a coach choose a team of 5 from 10
available athletes if
a) 2 specified athletes must be selected?
b) there are no restrictions?
24. In how many of the numbers between 1000 and 9999 does
the digit 3 occur?
25. How many "words", each of 2 vowels and 2 consonants, can
be formed from the letters of the word INVOLUTE?
26. How many diagonals are there in a 20-sided polygon?
27. A certain polygon has 35 diagonals ? How many sides does it
have?
are fowards, and 4 are guards. In how many ways can the
coach choose the starting lineup?
41. How many even 4-digits natural numbers are there?
28. A symphony is recorded on 4 CD's. In how many ways can
the CD's be played so that some part of the symphony is out
-of its correct-order?
42. If 6Pk =120, find k.
29. A railway coach has 10 seats facing forwards and 10 seats
facing backwards. In how many ways can 9 passengers be
seated if 2 refuse to ride facing forwards and 3 refuse to ride
facing backwards?
44. In how many ways can 4 identical pennies and 3 identical
nickels be arranged in a row?
30. From a company of 20 soldiers, a squad of 3 men is chosen
each night.
a) For how many consecutive nights could a squad go on
duty without any two of the squads being identical?
b)
In how many of these squads would Alphonse serve?
43. If P5 =15 120, find n.
45. In a group of 8 people, each person shakes hands with each
other person. How many handshakes are made?
46. If you must always move up or to the right, how many
different paths are there from A to B in each diagram?
a)
b)
B
31. How many 3-digit numbers are divisible by 5?
A
32. Solve „C2 = 45 for n.
A
33. If
6C3 + 6C4 = 7C
,, find the value of r.
10
34. Find the value of F,(i0Cr).
r=0
35. The central Alberta AA Bantam Hockey League consists of
11 teams. Each team plays each other team 3 times in the
season.
a) How many games will Cold Lake play?
b)
How many games must be scheduled for the league?
36. One of 2 parallel lines contains 5 marked points and the other
has 4 marked points. Using the marked points as vertices,
how many different
a)
b)
triangles can be formed?
quadrilaterals can be formed?
47. Expand and simplify (first 3 terms and the last):
b) (x2 - x )4
a) .{2x - 3y)5
48. Given the binomial power (x + 2)'0,
a) how many terms are in its expansion?
b) find the 41 term in its expansion.
c)
find the term of degree 3.
49. An airline pilot reported that in 7 consecutive days she spent,
in an unspecified order, 1 day in Winnipeg, 1 day in Regina,
2 days in Edmonton, and 3 days in Yellowknife. How many
different itineraries are possible if
a) there are no restrictions on order?
b) the first and last days were spent in Yellowknife?
37. An artist has 10 paintings. In how many ways can 8 or more
of these paintings be selected to be displayed at an art show?
50. If all the letters in the word DIPLOMA are used, how many,
different 7-letter arrangements can be made in which each
begins with 3 vowels?
38. Given the word SHUFFLE, how many "words" can be
formed using
51. A term in the binomial expansion of (ax + y)$, where a > 0, is
112x . What is the value of a?
a)
b)
all of the letters in each "word"?
4 of the letters in each "word"?
39. An exam has 20 multiple choice questions with 4 choices for
each question.
a) How many different answer keys could be formed?
b)
What is the probability that you could score 100% just
by guessing?
c)
What is the probability that you could score 50% just by
guessing?
52. Solve and verify your solution: ,,C2 = n3 3
53. In a group of people, there are 10 females and 12 males.
Determine the number of 4-member committees consisting of
at least 1 female that can be formed.
54. A teacher tells his students that on a multiple-choice test of
12 questions there are an equal number of A's, B's, C's, and
D's where each question has these 4 choices . How many
different answer keys are possible?
40. Ten players are competing for the 5 starting positions on a
school basketball team. Those positions are 1 center, 2
forwards, and.2 guards. Of those competing, 3 are centers, 3
55. Determine the constant term in the expansion of (2x2 - 1 )6
14
Math 30P
Permutations and Combinations Worksheet
la) T b) T c) F d) F e) T f) F g) F h) T i) T 2) 256 3) 120 4) 120 5a) 20 b) 81 6a) 17 576 000 b) 11 232 000 7) 360
8) 720 9) 340 10) 36 11) 792 12a) 92 378 b) 488 864 376 13) 120 14) 1716 15a) 2 598 960 b) 1 299 480 c) 454 480
15d) 131 560 16) 3003 17a) 720 b) 240 c) 480 18) 2450 19) 30 030 20a) 20 b) 15 21) 45 360 22a) 120 b) 240 c) 12
23a) 56 b) 252 24) 3168 25) 864 26) 170 27) 10 28) 23 29) 2 122 848 000 30a) 1140 b) 171 31) 180 32) 10 33) 3 or 4
34) 1024 35a) 30 b) 165 36a) 70 b) 60 37) 56 38a) 2520 b) 480 39a) 420 b) 1/420 c) 9.92 x 10-3 40) 54 41) 4500
42) 3 43) 9 44) 35 45) 28 46a) 70 b) 350 47a) 32x5 - 240x4y + 720x3y2 + ... - 243y5
47b) x8 12x5 + 54x2 + ... + X4 48a) 11 b) 960x' c) 15 360x3 49a) 420 b) 60 50) 144 51) 2 52) 5 53) 6820 54) 369 600
55) 60
Math 30P
Permutations and Combinations Worksheet
la) T b) T c) F d) F e) T f) F g) F h) T i) T 2) 256 3) 120 4) 120 5a) 20 b) 81 6a) 17 576 000 b) 11 232 000 7) 360
8) 720 9) 340 10) 36 11) 792 12a) 92 378 b) 488 864 376 13) 120 14) 1716 15a) 2 598 960 b) 1 299 480 c) 454 480
15d) 131 560 16) 3003 17a) 720 b) 240 c) 480 18) 2450 19) 30 030 20a) 20 b) 15 21) 45 360 22a) 120 b) 240 c) 12
23a) 56 b) 252 24) 3168 25) 864 26) 170 27) 10 28) 23 29) 2 122 848 000 30a) 1140 b) 171 31) 180 32) 10 33) 3 or 4
34) 1024 35a) 30 b) 165 36a) 70 b) 60 37) 56 38a) 2520 b) 480 39a) 420 b) 1/420 c) 9.92 x 10-3 40) 54 41) 4500
42) 3 43) 9 44) 35 45) 28 46a) 70 b) 350 47a) 32x5 - 240x4y + 720x3y2 +
243y5
47b) x$ - 12x5 + 54x2 +
+ 7 48a) 11 b) 960x7 c) 15 360x3 49a) 420 b) 60 50) 144 51) 2 52) 5 53) 6820 54) 369 600
55) 60
Math 30P
Permutations and Combinations Worksheet
la) T b) T c) F d) F e) T f) F g) F h) T i) T 2) 256 3) 120 4) 120 5a) 20 b) 81 6a) 17 576 000 b) 11 232 000 7) 360
8) 720 9) 340 10) 36 11) 792 12a) 92 378 b) 488 864 376 13) 120 14) 1716 15a) 2 598 960 b) 1 299 480 c) 454 480
15d) 131 560 16) 3003 17a) 720 b) 240 c) 480 18) 2450 19) 30 030 20a) 20 b) 15 21) 45 360 22a) 120 b) 240 c) 12
23a) 56 b) 252 24) 3168 25) 864 26) 170 27) 10 28) 23 29) 2 122 848 000 30a) 1140 b) 171 31) 180 32) 10 33) 3 or 4
34) 1024 35a) 30 b) 165 36a) 70 b) 60 37) 56 38a) 2520 b) 480 39a) 420 b) 1/420 c) 9.92 x 10-3 40) 54 41) 4500
42) 3 43) 9 44) 35 45) 28 46a) 70 b) 350 47a) 32x5 - 240x4y + 720x3y2 +
243y5
47b) x8 - 12x5 + 54x2 + ,,, + 84 48a) 11 b) 960x7 c) 15 360x3 49a) 420 b) 60 50) 144 51) 2 52) 5 53) 6820 54) 369 600
55) 60
Math 30P
Permutations and Combinations Worksheet
la) T b) T c) F d) F e) T f) F g) F h) T i) T 2) 256 3) 120 4) 120 5a) 20 b) 81 6a) 17 576 000 b) 11 232 000 7) 360
8) 720 9) 340 10) 36 11) 792 12a) 92 378 b) 488 864 376 13) 120 14) 1716 15a) 2 598 960 b) 1 299 480 c) 454 480
15d) 131 560 16) 3003 17a) 720 b) 240 c) 480 18) 2450 19) 30 030 20a) 20 b) 15 21) 45 360 22a) 120 b) 240 c) 12
23a) 56 b) 252 24) 3168 25) 864 26) 170 27) 10 28) 23 29) 2 122 848 000 30a) 1140 b) 171 31) 180 32) 10 33) 3 or 4
34) 1024 35a) 30 b) 165 36a) 70 b) 60 37) 56 38a) 2520 b) 480 39a) 420 b) 1/420 c) 9.92 x 10-3 40) 54 41) 4500
42) 3 43) 9 44) 35 45) 28 46a) 70 b) 350 47a) 32x5 - 240x4y + 720x'y2 +
243y5
47b) x$ - 12x5 + 54x2 + ... + 84 48a) 11 b) 960x' c) 15 360x3 49a) 420 b) 60 50) 144 51) 2 52) 5 53) 6820 54) 369 600
55) 60
Math 30P
Permutations and Combinations Worksheet
la) T b) T c) F d) F e) T f) F g) F h) T i) T 2) 256 3) 120 4) 120 5a) 20 b) 81 6a) 17 576 000 b) 11 232 000 7) 360
8) 720 9) 340 10) 36 11) 792 12a) 92 378 b) 488 864 376 13) 120 14) 1716 15a) 2 598 960 b) 1 299 480 c) 454 480
15d) 131 560 16) 3003 17a) 720 b) 240 c) 480 18) 2450 19) 30 030 20a) 20 b) 15 21) 45 360 22a) 120 b) 240 c) 12
23a) 56 b) 252 24) 3168 25) 864 26) 170 27) 10 28) 23 29) 2 122 848 000 30a) 1140 b) 171 31) 180 32) 10 33) 3 or 4
14) 1024 35a) 30 b) 165 36a) 70 b) 60 37) 56 38a) 2520 b) 480 39a) 420 b) 1/420 c) 9.92 x 10-3 40) 54 41) 4500
42) 3 43) 9 44) 35 45) 28 46a) 70 b) 350 47a) 32x5 - 240x4y + 720x3y2
243y5
47b) x8 - 12x5 + 54x2 + ,,, + 84 48a) 11 b) 960x' c) 15 360x3 49a) 420 b) 60 50) 144 51) 2 52) 5 53) 6820 54) 369 600
55) 60