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Pathways and Pascal's Triangle

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Pathways and Pascal's Triangle

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Chapter 7 Combinatorics

7.4

Pathways and

Pascal's Triangle

7.4.1

MATHPOWERTM 12, WESTERN EDITION

Pascal’s Triangle

Pascal’s triangle is an array of natural numbers. The sum of any

two adjacent numbers is equal to the number directly below them.

Sum of

each row

1

1

20

1st Row

2

1

21

2nd Row

1

4

1

22

3rd Row

1

2

1

23

8

4th Row

3

3

1

16

4

24

1

6

4

5th Row

1

32

1

25

5

10

1

10

5

6th Row

26

64

1

6

15

15

20

6

1

7th Row

128

35

21

7

1

27

35

7

21

1

8th Row

nth Row

2n - 1

7.4.2

Pathways and Pascal’s Triangle

Pascal’s triangle can be used to solve pathway problems.

Pascal’s Triangle

A

1

1

A

C

1

2

1

B

1

B

D

1 1

There is only 1 path

from A to C and only

1 path from A to D.

There are 2 paths

from A to B.

1 2 1

Again, this relates to

Pascal’s triangle.

1 3 3 1

This relates to

Pascal’s triangle.

1 4 6 4 1

1

1

A

Use Pascal’s triangle to

connect the corners of each

square for each sum.

2

3

1

3

6

1

B

7.4.3

Pathways and Pascal’s Triangle

Continue with the pattern of Pascal’s triangle

to solve larger pathway problems.

1

1

1

1

1

1

1

1

A

A

4

2

5

3

2

4

5

3

1

1

15

6

10

3

3

6

10

15

1

1

B

4

20

10

35

1

15

35

5

70

B

1

To simplify these problems, you can use combinatorics:

This grid has 4 squares across

and 4 squares down.

This grid has 4 squares across and 2 squares down.

8C4

= 70

6C2 = 15

7.4.4

Pathways and Pascal’s Triangle

Determine the number of pathways from A to B.

A

A

1.

2.

B

B

10C5x8C3 = 14 112

14C3x5C3 x8C2 = 101 920

7.4.5

Pathways --An Application

In a television game show, a network of paths into which a ball

falls is used to determine which prize a winner receives.

a) How many different paths are there to each lettered slot?

b) What is the total number of paths from top to bottom?

There is only one pathway to each of Slots A and F.

There are five pathways to each of Slots B and E.

There are ten pathways to each of Slots C and D.

The total number of pathways from top to bottom

is 32.

(Row 6 of Pascal’s triangle, n = 5: 25 = 32)

1

5

5

10

1

10

The total number of pathways

from top to bottom is 128.

Determine the number of

pathways from top to

bottom for this network.

(Row 8 of Pascal’s triangle,

n = 7: 27 = 128)

7.4.6

Assignment

Suggested Questions

Pages 352 and 353

1-4, 7, 8

7.4.7