Transcript PASCAL’S TRIANGLE

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.
ROW 0
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*
ROW 1
3. The basic idea, is to add the top two digits, and
its sum goes in the box below.
ROW 3
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 2
The Formula

Another way of
calculating the
numbers is by using
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
Recursive Formula: tn,r = tn-1,r-1 + tn-1,r
Using the recursive formula how do we figure out what
row n = 9?
1.
First let tn= t9,1
*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.
2. t 9,1= t 9 -1,1-1+ t 9 -1,1
= t8,0+ t8,1
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
Continued
Find: row 8 position 0 and row 8 position 1 and we add them.
row 8
position 1
row 8
position 0
=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 5 = sum of 32
Row 6 = sum of 64
Row 7 = sum of 128
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)
Example 3
* log a^b= b log a
Which 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.
Try it….
Try finding a pattern?
1.
2.
Try identfying a pattern with
PASCAL's triangle on your
own.
Here is a website with some
interesting patterns.

3.
http://ptri1.tripod.com
For further practice read the
key concepts on page 251.
Sierpinski Triangle
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.



Applying Pascals Methods Cont’d
The water has one path to each
of the buckets in the second
row. There is one path to each
outer bucket of the third row,
but two paths to the middle
buckets, and so on.
The number in the diagram
match those in Pascal's triangle
because they were derived
using the same methodPascal's methods.
*NOTE:
 Pascal's method involves
adding two neighboring
terms in order to find the
term below.
 Pascal's method can be
applied to counting paths in
a variety of arrays and grids.
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?
*Note: As
you keep
branching
out, the out
extreme
values
continue to
be equal to 1
on both sides
(P-C).
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)
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.
Therefore the answer
is 5 + 9 + 8 + 8 = 30
3. The square containing an X gets a zero or no number
since there are no paths through this blocked square.
Example 3: Fill in the Missing Numbers
792
……..
924
……..
1716
……..
1287
………
3003
330
……..
495
5115
………
825
2112
1. Firstly, considering row number 1 from the bottom, it is the addition
2.
3.
of the previous consecutive 2 terms (3003 and 2112).
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;
(Let the missing term be x.)
Example 3: Fill in the Missing Numbers
Cont’d
792
……..
924
……..
1716
……..
1287
………
3003
5115
………
4. According to Pascal’s method,
 X+825=2112
 X=2112-825
 X=1287
330
……..
495
825
2112
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).
5. This procedure can be followed and implied to solve the rest of
the table