PASCAL’S TRIANGLE

History and Creation Pascal Blaise was a French

mathematician, physicist and a religious philosopher.

His famous contribution was the Pascal’s Triangle, although Pascal wasn’t the first mathematician to study this triangle.

With the help of the Triangle, Pascal was able to solve problems in probability.

How does it work?

1. To begin, we start at the top, row 0, by placing the first digit, 1.

2. Before we calculate, since there is no number beside the first box, we carry the 1 down or imagine a 0 next to the first and last box. *This applies to every row*

3. The basic idea, is to add the top two digits, and its sum goes in the box below.

4. Then , the 2nd and 3rd digit is added and the sum goes in the box below them both.

5. This step goes on until the row is completed, then we go to the next row and begin again.

ROW 0

ROW 1

ROW 2

ROW 3

The Formula Another way of Another way of

calculating the calculating the numbers is by using numbers is by using the recursive formula: the recursive formula: tn,r = tn-1,r-1 + tn-1,r

n = the row number.r = position on the row.

Row 0

Row 1

Row 2

Row 3

Row 4

Row 5

Row 6

Example 1

Using the recursive formula how do we figure out what

row n = 9?

1. First let tn= t9,1

2. t 9,1= t 9 -1,1-1+ t 9 -1,1

= t8,0+ t8,1

*NOTE: We write tn= t9,1 because we are trying to figure out row 9 and since the position we are trying to figure out in row 9 is 1 we place 1 in the r, r =1.

Recursive Formula: tn,r = tn-1,r-1 + tn-1,r

3. Then we go to Pascal’s triangle for reference and find row 8 position 0 and row 8 position 1 and we add them.

Next pg

ContinuedFind: row 8 position 0 and row 8 position 1 and we add them.

row 8 position 0

row 8 position 1 =1+8

= 9

Example 2

Determine the rule relating to the row number and the sum of that row?

1. First thing is to add all the numbers in the first row and continue until you have data.

2. Row 0 = sum of 1

Row 1 = sum of 2

Row 2 = sum of 4

Row 3 = sum of 8

Row 4 = sum of 16

Find any pattern yet…….?

3. The answer to the question is 2n ( n represents the row number)

Row 5 = sum of 32

Row 6 = sum of 64

Row 7 = sum of 128

Example 3Which row in Pascal’s triangle has the sum of 1024?

1. Since, 2n is the sum of all the numbers in any row. Using the ‘log’ button on your calculator is an easy way to solve this.

2. using information from the question, we can derive the following equation

3. 2^n= 1024

4. Taking the log of both sides, we get log2^n=log1024 *

5. n=log1024 / log2

6. n=10

7. Therefore, row 10 has the sum of 1024.

* log a^b= b log a

Try it….

1. Try identfying a pattern with PASCAL's triangle on your own.

2. Here is a website with some interesting patterns.

http://ptri1.tripod.com

3. For further practice read the key concepts on page 251. Sierpinski Triangle

Try finding a pattern?

Applying Pascals Methods

The iterative process that generates the terms in Pascal's triangle can also be applied in real life to even the simplest of things such as counting the number of paths or routes between two points.

You can use Pascal's method to count the different paths that water overflowing from the top bucket could take to each of the buckets in the bottom row.

The water has one path to each The water has one path to each of the buckets in the second of the buckets in the second row. There is one path to each row. There is one path to each outer bucket of the third row, outer bucket of the third row, but two paths to the middle but two paths to the middle buckets, and so on. buckets, and so on.

The number in the diagram The number in the diagram match those in Pascal's triangle match those in Pascal's triangle because they were derived because they were derived using the same method-using the same method-Pascal's methods.Pascal's methods.

*NOTE:*NOTE: Pascal's method involves Pascal's method involves

adding two neighboring adding two neighboring terms in order to find the terms in order to find the term below.term below.

Pascal's method can be Pascal's method can be applied to counting paths in applied to counting paths in a variety of arrays and grids.a variety of arrays and grids.

Applying Pascals Methods Cont’d

Example 1: Counting Paths In an Array. Determine how many different paths will spell PASCAL if

you start at the top and proceed to the next row by moving diagonally left or right?

1. Start from P. You can either go to the left A or to the right A. (1 path)

2. There is one path from an A to the left S, two paths from an A to the middle S and one path from an A to the right S.

3. Continuing with this counting reveals that there are 10 different paths leading to each L. Therefore, a total of 20 paths to spell PASCAL. (10+10=20)

*Note: As you keep branching out, the out extreme values continue to be equal to 1 on both sides (P-C).

Example 2: Counting paths on a Checkerboard.On the checkerboard shown, the checker can travel only diagonally upward. It cannot move through a square containing an X. Determine the number of paths the checker's current position to the top of the board?

1. There is one path possible into each of the squares diagonally adjacent to the checker's starting position.

2. From the second row, there are 4 paths to the third row: one path to the 3rd square from the left, two to the 5th square and one to the 7th square.

3. The square containing an X gets a zero or no number since there are no paths through this blocked square.

Therefore the answer is 5 + 9 + 8 + 8 = 30

Example 3: Fill in the Missing NumbersExample 3: Fill in the Missing Numbers

…….. …….. 495 …….. …….. ……… 825 3003 2112 ………

924 7921716

5115

1287330

1. Firstly, considering row number 1 from the bottom, it is the addition of the previous consecutive 2 terms (3003 and 2112).

2. In row 3, the value 1287(2nd term) is derived from subtracting the term beside it and the term is the next row below it. It can be worked out as follows;

3. (Let the missing term be x.)

Example 3: Fill in the Missing Numbers Example 3: Fill in the Missing Numbers Cont’dCont’d

…….. …….. 495 …….. …….. ……… 825 3003 2112 ………

924 7921716

5115

1287330

4. According to Pascal’s method, X+825=2112 X=2112-825 X=1287

5. This procedure can be followed and implied to solve the rest of the table

Note: Always begin solving for missing terms in places were there are at least 2 terms, i.e, 2 terms are known (As given in example above).