Download 1.11 Pascal`s Triangle.notebook

1.11 Pascal's Triangle.notebook
September 20, 2013
1.11 Pascal's Triangle
Position 1
Position 0
Row 0
Pascalʹs Triangle is built with an iterative process. Each term is equal to the sum of the two terms directly above it. The first and the last term in each row are both equal to 1.
Row 1
Row 2
Row 3
Row 4
t3,0
t3,1
t4,1
t0,0
t1,0
t2,0
t3,0
t1,1
t2,1
t3,1
Notation:
t2,2
t3,2
etc...
t3,3
termrow,position
tn,r
1
1.11 Pascal's Triangle.notebook
September 20, 2013
Pascal's Identity
The following relationship, known as Pascal's Identity, is often useful.
tn,r represents the term in row n, position r.
Thus, tn,r = tn­1,r­1 + tn­1,r
Example 1
Express as a single term from Pascal's Triangle.
a. t6,3 + t6,4 b. t14,5 + t14,6 c. t21,7 ‐ t20,6 d. t15,10 ‐ t14,10
Example 2
The 1st six numbers in row 25 are 1, 25, 300, 2300, 12650, 53130.
Determine the first six numbers of row 26.
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1.11 Pascal's Triangle.notebook
September 20, 2013
Row Sums
The sum of the terms in
any row n in Pascal's
Triangle is 2 n.
eg. sum of row 9 = 29 = 512
eg. row whose sum is 4096 is row
12, ∵ 212 = 4096
Example 3
What is the sum of row 7?
Example 4
Which row in Pascal's triangle has a sum of 32,768?
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1.11 Pascal's Triangle.notebook
September 20, 2013
Combinations
Is there any link between Pascalʹs triangle and Combinations?
Recall:
t0,0
n
( )
= nCr = C(n,r)
r
t1,0
t2,0
t3,0
t1,1
t2,1
t3,1
t2,2
t3,2
t3,3
etc...
Example 5
Determine the first 5 numbers of the 18th row of Pascal's triangle.
Example 6
The 6th and 7th term of a row is 792 & 924. Which row?
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1.11 Pascal's Triangle.notebook
September 20, 2013
Triangle Numbers
Coins can be arranged in the shape of an equilateral triangle.
Determine the number of coins in the triangle when there are
four, five and six rows.
Locate these numbers in Pascal's triangle to the number of coins in a
triangle with n rows.
Example 7
How many coins are there in a triangle with 12 rows?
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1.11 Pascal's Triangle.notebook
September 20, 2013
11 and Pascal
Relate 11 to Pascals' Triangle
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1.11 Pascal's Triangle.notebook
September 20, 2013
Applying Pascal's Method
Pascal's Triangle is often used to determine the number of paths between points.
Example 8
Determine how many different paths will spell SUCCESS if you start at
the top and proceed to the next row moving diagonally left or right.
S
U U
C C C
C C C C
E E E
S S
S
Example 9
In Plinko, a token slides down a board. If the token cannot go through a
shaded square, in how many ways could you win $500?
START
$500 $500 7
1.11 Pascal's Triangle.notebook
September 20, 2013
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