Pascal's Method can be applied to counting paths or routes between two points in a variety of arrays and grids.

Pascal's Triangle:



Where do we use Pascal's triangle:
Pascal's Triangle is more than just a triangle filled with numbers. Many times, one does not know what to do with these numbers, or even how to use them. However, there are two major areas where these numbers in the Pascal Triangle can be used, and these areas are Algebra and Probability/Combinations.

PROBABILITY/COMBINATIONS:

The Pascal triange can be used to find combinations.

Example:
Jon has five hats on a rack, and he wants to know how many different ways he can pick two of them and wear them. It doesn't matter which hat is on top, it just matters which two hats he pick. This problem leads to the question "how many different ways can you pick two objects from a set of five objects?"
The answer is the number in the second place in the fifth row, i.e. 10. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1. Due to choosing this property, the binomial coefficient [6:3] is usually read "six choose three." If you want to find out the probability of choosing one particular combination of two hats, then that probability is 1/10.

For further explanation on how Pascal's triangle works:


Each row added is equivalent to the sum of numbers in that row.
That sum can be multiplied by 2 to get the sum of the next row.
Example: 1x2=2... 2x2=4... 4x2=8... 8x2=16... etc.

A Video to further explain Pascal's Triangle:



Here is an examlpe from 4.5 Homework:
Page 256:

2) In the following arrangements of letters, start from the top and proceed to the next row by moving diagonally left or right. How many different paths will spell each word?


Therefore, there are 128 different ways of spelling each word.


Links to other explanations to Pascal's triangle:
http://ptri1.tripod.com/
http://www.krysstal.com/binomial.html
To create a Pascal's Triangle of your own, click here:
http://www.cecm.sfu.ca/organics/papers/granville/support/pascalform.html

References
Mathematics of Data Management , Grade 12, (MDM4U). McGraw-Hill Ryerson