1. Pathways and
Pascal's Triangle

2. 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 1 1 20
2nd Row 1 1 2 21
3rd Row 1 2 1 4 22
4th Row 1 3 3 1 8 23
5th Row 1 4 6 4 1 16 24 1 32 25
6th Row 1 5 10 10 5 26 1 6 15 20 15 6 1 64
7th Row 1 7 21 35 35 21 7 1
128 27
8th Row
nth Row
2n - 1

3. Pathways and Pascal’s Triangle
Pascal’s triangle can be used to solve pathway problems. A
Pascal’s Triangle A 1 1 C 1 2 1 1 B D B
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.
Again, this relates to
Pascal’s triangle.
This relates to
Pascal’s triangle. A 1 1
Use Pascal’s triangle to
connect the corners of each
square for each sum. 2 3 1 3 6 1 B

4. 8C4 = 70 6C2 = 15 Pathways and Pascal’s Triangle
Continue with the pattern of Pascal’s triangle
to solve larger pathway problems. 1 1 1 1 1 1 A 1 A 1 2 3 4 5 2 3 4 5 1 1 3 6 10 15 3 6 10 15 1 1 B 4 10 20 35 1 5 15 35 70 1 B
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

5. Pathways and Pascal’s Triangle
Determine the number of pathways from A to B. A A 1. 2. B B
10C5 x 8C3 =
14C3 x 5C3 x 8C2 =

6. 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 10 10 5 1
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)