Chapter 7 Combinatorics
7.4
MATHPOWERTM 12, WESTERN EDITION 7.4.1
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
1st Row
2nd Row
3rd Row
4th Row
5th Row
6th Row
7th Row
8th Row
nth Row
7.4.2
Pathways and Pascal’s Triangle
Pascal’s triangle can be used to solve pathway problems.
A
D
C
B
Use Pascal’s triangle to
connect the corners of each
square for each sum.
Pascal’s Triangle
A
1
B
1
1
A
B
7.4.3
Pathways and Pascal’s Triangle
Continue with the pattern of Pascal’s triangle
to solve larger pathway problems.
A
A
B
B
To simplify these problems, you can use combinatorics:
7.4.4
Pathways and Pascal’s Triangle
Determine the number of pathways from A to B.
1.
A
2.
A
B
B
7.4.5
Pathways with holes.
Determine the number of pathways from A to B.
Note that you must always move closer to B.
A
1.
There are
paths from A to B.
B
A
2.
There are
paths
from
A to B.
B
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?
Determine the number of
pathways from top to
bottom for this network.
7.4.6
Suggested Questions
Pages 352 and 353
1-4, 7, 8
7.4.7