perms, comb, path, prob, bt review _ key
DESCRIPTION
permutations, combinations, probabilityTRANSCRIPT
Math 30P Permutations and Combinations Review
Perms, Comb, Path, Prob, BT Review.doc
Name: ________________
Math 30P Perms, Combs, Paths, Prob, Bi. Thm Review
1.The number of distinguishable arrangements of the letters of the word TEETER, taken all
at once is
2.nC2 =
3.The number of triangles that can be formed using six distinct points on the circumference of a
circle is
4.The number of ways that 5 people can be seated on 7 chairs is 7 ( 6 ( 5 ( 4 ( 3 or 7P5 = 25205.The solution of the equation 4(n ( 2C2) = nC3 is
6.The constant term in the expansion of is: Constant term means x0
7.The number of terms in the binomial expansion is 11, then the value of k is
Remember, the exponent is one less than the number of terms.
If there are 11 terms, the exponent must equal 10.
8.If a football league consists of 10 teams, the number of league games played during the
season in which each team plays exactly two games with each of the other teams is
This represents the number of games played if they played each other once.
9.The numerical coefficient of the third degree term in the expansion of (2x 1)5 is
10.Three boys and 4 girls are arranged randomly in a row. What is the probability that the
three boys are all separated?
Arrange the 4 girls first (4!) and then place the boys _ G _ G _ G _ G _
between the girls (5 ( 4 ( 3).
4!( (5 ( 4 ( 3) = 1440Total number of arrangements = 7! = 5040
Probability =
11.A box contains 8 light bulbs, 3 of which are defective. When selecting 3 light bulbs what is the
probability that you will select 3 good bulbs?
Probability =
12.You are one of a group of 12 people. A committee of 3 is randomly selected from this group. What is the probability that you will NOT be on the committee?
13.a. Find the number of different hands of 5 cards containing two aces and three kings which can be
dealt from a standard deck of 52 cards.
b. What is the probability of drawing 5 cards from a standard deck of cards and getting 2 aces and
3 kings?
14.If a regular polygon has 44 diagonals, find the number of sides.
11 sides
15.Find the number of 5 card hands from a deck of 52 cards that have at least 2 aces.
2 aces and 3 other cards or 3 aces and 2 other cards or 4 aces and 1 other card
16.The number of vanity automobile license plates which can be made by using 6 letters followed by
2 digits is given by n x 1010. The value of n is _______(nearest 100th and repetitions are allowed)
266 ( 102 = 3.089 ( 1010 n = 3.0917.Find the 7th term in the expansion of .k = 6
18.Fred has one of each of the following bills $5, $10, $20, $100. How many different sums of money can he have using one or more bills?
He can use 1, 2, 3 or 4 bills. or 24 ( 4C0 = 1519.A student must walk 5 blocks north and 3 blocks east to get to school. How many different routes
can the student take to get to school?
The different arrangements of N N N N N E E E will give the different routes.
20.In how many ways can 5 girls and 4 boys be arranged on a bench, if no two girls can sit together?
arrange the boys first (4!) and then place the girls between the boys (5 ( 4 ( 3 ( 2 ( 1).
(4!)(5 ( 4 ( 3 ( 2 ( 1) = 2880
21.How many different ways can you get from point A to point B in each of the following?
a)
b)
or You have to go RRDDF (2 right, 2 down and 1 forward)
They can be arranged ways
c)
Any time every path goes through the same point (M as in the above),
you can find how many ways to get to the point (M) and then how many
ways to get from the point to the end and multiply both. 56 ( 6 = 336
** Anytime a pathway has no irregularities you can determine the number of different paths
algebraically.
Question C: First rectangle RRRRRDDD can be arranged
Second rectangle RRDD can be arranged
Total number of paths = 56 ( 6 = 336
d) What is the probability that the path in 21 c) went through C?
Since all paths pass through C, you can multiple 10 ( 6 to get the number of pathways.
Probability =
22.Expand (2a + b)5
23.a) In how many ways can the letters of the word PERSON be arranged if the letters P and N must
be kept together?
Treat the two letters as one and then rearrange the two letters. 5! ( 2! = 240
b) What is the probability that SON appears in the arrangement?
Treat SON as one letter but do not rearrange them. 4! = 24
Probability =
24.How many numbers greater than 30 000 can be made from the digits 2, 3, 4, 5, 6 and 7
(Remember-Listed symbols can only be used as many times as they are listed)?
Can be any 6-digit number or any 5-digit number 6! + 5 ( 5 ( 4 ( 3 ( 2 = 1320
that begins with 3, 4, 5, 6, or 7.
25. A raffle sells 40 tickets and gives away 3 identical prizes (A graphing calculator!). A ticket can win only one prize. If you purchase 5 tickets, what is the probability that:
a) you win no prize?
b) you win exactly one prize?
Select 3 tickets from the 35 that you do Select one of your 5 tickets and 2 from the other 35.
not have.
26.Simplify the following.
a)
27. The letters of the word HEART are arranged in a row at random. What is the probability that:
Total number of arrangements = 5! = 120
a) the two vowels are together?
b) the word ART appears in the arrangement?
c) the vowels are at the ends?
d) it starts with a vowel?
e) the consonants are separated?
28. The letters B, B, B, C, D, E, and O are arranged (all at once) in a row at random. What
is the probability that:
Total number of arrangements =
a) the 3 B's are together?
b) the 3 B"s are separated?
c) the first letter is B and the second letter is not B?
d) the first and last letters are vowels?
29. A committee of 5 is selected from 5 girls and 9 boys. What is the probability that the
committee:
Total number of committees = 14C5 = 2002
a) 2 girls and 3 boys?
b) including a specific girl and excluding 2 specific boys?
c) all boys.
d) having at least 2 boys?
This is doing it: total ( 0 boys 5 girls ( 1 boy 4 girls
You can also do it 2 boys 3 girls + 3 boys 2 girls + 4 boys 1 girl+ 5 boys 0 girlsAnswers:
1) b 2) b 3) b 4) a 5) b 6) c 7) a 8) b 9) d 10) b 11) b 12) a
13 a) 24 b) 14) 11 15) 108 336 16) 3.09 17) 672 18) 15
19) 56 20) 2880 21 a) 30 b) 64 c) 336 d)
22) 32a5 + 80a4b +80a3b2 + 40a2b3 + 10ab4 + b523 a) 240 b) 24) 1320
25 a) b) 26) 5987 b) 27) a) b) c) d)
e) 28 a) b) c) d) 29 a) b) c) d)
C
B
A
30
6
12
6
3
12
1
3
3
3
2
2
1
2
1
1
1
B
A
64
30
10
34
20
10
4
14
10
6
3
4
4
3
1
1
1
1
2
1
1
B
A
1
1
1
1
1
1
1
1
2
3
4
5
6
3
6
10
15
21
4
10
20
35
56
56
56
56
112
168
56
168
336
(
M
1
1
1
2
1
3
3
6
B
M
A
B
C
1
1
2
3
1
1
30
1
3
6
4
10
10
10
10
10
10
20
30
60
b) EMBED Equation.DSMT4
1
12
_1265709295.unknown
_1265886532.unknown
_1265900016.unknown
_1265901795.unknown
_1271440688.unknown
_1271440701.unknown
_1271440302.unknown
_1265901805.unknown
_1265901772.unknown
_1265901784.unknown
_1265901761.unknown
_1265894627.unknown
_1265899660.unknown
_1265899695.unknown
_1265899715.unknown
_1265899676.unknown
_1265899621.unknown
_1265888158.unknown
_1265894280.unknown
_1265894357.unknown
_1265894230.unknown
_1265887042.unknown
_1265728712.unknown
_1265731796.unknown
_1265886410.unknown
_1265729665.unknown
_1265729920.unknown
_1265729974.unknown
_1265729772.unknown
_1265729205.unknown
_1265712474.unknown
_1265728352.unknown
_1265712406.unknown
_1098697220.unknown
_1265708147.unknown
_1265708371.unknown
_1265708668.unknown
_1265708885.unknown
_1265708442.unknown
_1265708266.unknown
_1265708342.unknown
_1265708224.unknown
_1265699114.unknown
_1265708091.unknown
_1265708121.unknown
_1265708071.unknown
_1265701517.unknown
_1226318535.unknown
_1265698931.unknown
_1113307532.unknown
_1098680443.unknown
_1098680542.unknown
_1098690640.unknown
_1098697219.unknown
_1098680702.unknown
_1098680727.unknown
_1098680890.unknown
_1098680683.unknown
_1098680510.unknown
_1098680525.unknown
_1098680488.unknown
_1098680353.unknown
_1098680438.unknown
_1098680439.unknown
_1098680437.unknown
_1098679012.unknown
_1098680028.unknown
_1098679161.unknown
_1098620757.unknown
_1098621412.unknown
_1098617781.unknown