Math 30P Permutations and Combinations Review

Perms, Comb, Path, Prob, BT Review.doc

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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

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