4.1 Organized Counting

McGraw-Hill Ryerson Mathematics of Data Management, pp. 225–231

1. Draw a tree diagram to illustrate the possible travel itineraries for Pietro if he can travel from home to Ottawa by bus, car, or train, and then from Ottawa to Orlando, Florida by bus, train, or plane.

2. A new computer game has two possibilities, the gate is either open or closed. The characters must meet five gates during their time allotment. Draw a tree diagram to illustrate the possible scenarios for a player.

3. A physical education teacher has five pairs of running shoes, eight pairs of sweat pants, and 12 T-shirts. How many different outfits cans she wear?

4.2 Factorials and Permutations

McGraw-Hill Ryerson Mathematics of Data Management, pp. 232–240

1. Mrs. Edwards has to mark 33 quizzes tonight. In how many different orders can she mark them?

2. In how many ways can you choose a president and vice-president from a group of 11 people?

3. A quarterback has a series of six plays possible. If the coach asks the quarterback not to repeat any plays in a game, how many different orders of plays is possible?

4. Assuming that everyone in a particular school has three initials, find out what is the smallest number of students in a school for which there must be at least two with the same initials.

5. Use a calculator to find each of the following:

a) 5P2 b) 10P10 c) 8P1 d) 15P7

4.3 Permutations With Some Identical Elements

McGraw-Hill Ryerson Mathematics of Data Management, pp. 241–246

1. In how many different ways can your arrange five flags in a row if there are two red flags and three blue flags?

2. How many distinct "words" can you make using all the letters of the word PARALLELEPIPED?

3. In how many ways can you arrange all the letters of the word TORONTO

a) that begin with a T?

b) that end in a T?

c) that have both Ts together?

4. Find the number of ways of arranging all of the letters of the word TENNESSEE

a) if there are no restrictions

b) if the first two letters must be EE

c) if the first two letters must not be EE

4.4 Pascal's Triangle

McGraw-Hill Ryerson Mathematics of Data Management, pp. 247–253

 

1. Fill in the missing numbers of this part of Pascal’s triangle.
 

36

 		 

84

 		 

___

 		 

126
   

___

 		 

210

 		 

___

 		 
     

___

 		 

___

 		   

 

 

2. Find the missing numbers of this part of Pascal’s triangle.
 

___

 		 

25

 		 

300

 		 

___
   

26

 		 

___

 		 

2600

 		 
     

___

 		 

___

 		   

 

3. Determine the sum of the terms in the 10th row of Pascal’s triangle.

3. In the following arrays of letters, start at the top and proceed to the next row diagonally left or right. How many different paths will spell each word?
 

a)

 		     

P

 		       

b)

 		     

F

 		     
     

A

 		 

A

 		         

E

 		 

R

 		   
   

S

 		 

S

 		 

S

 		     

R

 		 

R

 		 

R

 		 
 

C

 		 

C

 		 

C

 		 

C

 		 

M

 		 

M

 		 

M

 		 

M
   

A

 		 

A

 		 

A

 		     

A

 		 

A

 		 

A

 		 
     

L

 		 

L

 		         

T

 		 

T

 		   

c)

    

S

 		     

d)

 		   

G

 		   
     

I

 		 

I

 		     

A

 		 

A

 		 
   

E

 		 

E

 		 

E

 		 

U

 		 

U

 		 

U
     

R

 		 

R

 		     

S

 		 

S

 		 
   

P

 		 

P

 		 

P

 		     

S

 		   
 

I

 		 

I

 		 

I

 		 

I

 		         
   

N

 		 

N

 		 

N

 		           
 

S

 		 

S

 		 

S

 		 

S

 		         
   

K

 		 

K

 		 

K

 		           
     

I

 		 

I

 		             

 

4. Determine the number of possible routes from X to Y if you travel only north or west.

 

a)

Y

 		             

b)

Y

 		       
                               
                               
                               
               

X

 		             
                               
                             

X

 

ANSWERS

1.

Bus

Car

Train
       
 

Bus Train Plane

Bus Train Plane

Bus Train Plane

2.

open

open

open

 

 

 

closed

open

 

closed

open

closed
 

open

closed
 

open

 

closed

open

closed
 

open

closed
 

closed

open

 

 

 

closed

open

 

closed

open

closed
 

open

closed
 

open

 

closed

open

closed
 

open

closed
 
 

closed

open

open

 

 

 

closed

open

 

closed

open

closed
 

open

closed
 

open

 

closed

open

closed
 

open

closed
 

closed

open

 

 

 

closed

open

 

closed

open

closed
 

open

closed
 

open

 

closed

open

closed
 

open

closed

3. 480

4.2 Factorials and Permutations

1. 33!

2. 110

3. 720

4. 17 577

5. a) 20 b) 3 628 800 c) 8 d) 32 432 400

4.3 Permutations With Some Identical Elements

1. 10

2. = 201 801 600

3. a) 120 b) 120 c) 120

4. a) 3780 b) 630 c) 3150

 

4.4 Pascal's Triangle

1. 126, 120, 252, 330, 462

2. 1, 2300, 325, 351, 2925
  1. 1024

2. a) 1, 100, 4950, 161700

b) 161 700, 4950, 100, 1

c) 161 700x3y97, 4950x2y98, 100xy99, y100

3. a) 20 b) 20 c) 204 d) 6
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