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PASCAL’ S T RIANGLE
Todd Cochrane
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Robert’s Dream
In his dream, Robert and the Number Devil build a giant
pyramid of blocks. Being two-dimensional they agree to call it a
triangle of blocks. Then they start marking the blocks with a felt
pen, labeling the top block 1, the next two 1,1, and then each
block thereafter the sum of the two values above it. Thus, the
third row is 1,2,1, the fourth 1,3,3,1 and so on.
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Pascal’s Triangle
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
1 8 28 56 70 56 28 8 1
Rule: Each term in Pascal’s triangle is the sum of the two
terms above it.
Pascal’s triangle is named after Blaise Pascal, who put together
many of its properties in 1653.
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Diagonal’s of Pascal’s Triangle
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
1 8 28 56 70 56 28 8 1
Diagonals: Note, the Southwesterly and Southeasterly
diagonals are the same.
Diagonal 1:
1, 1, 1, 1, . . .
Diagonal 2:
1, 2, 3, 4, 5, . . . ...
(yawn)
Natural Numbers
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Diagonal’s of Pascal’s Triangle
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
1 8 28 56 70 56 28 8 1
Diagonals: Note, the Southwesterly and Southeasterly
diagonals are the same.
Diagonal 1:
1, 1, 1, 1, . . .
Diagonal 2:
1, 2, 3, 4, 5, . . . ...
(yawn)
Natural Numbers
Diagonal 3: 1, 3, 6, 10, 15, 21, . . .
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Diagonal’s of Pascal’s Triangle
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
1 8 28 56 70 56 28 8 1
Diagonals: Note, the Southwesterly and Southeasterly
diagonals are the same.
Diagonal 1:
1, 1, 1, 1, . . .
Diagonal 2:
1, 2, 3, 4, 5, . . . ...
Diagonal 3: 1, 3, 6, 10, 15, 21, . . .
(yawn)
Natural Numbers
Triangle Numbers.
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Diagonal’s of Pascal’s Triangle
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
1 8 28 56 70 56 28 8 1
Diagonals: Note, the Southwesterly and Southeasterly
diagonals are the same.
Diagonal 1:
1, 1, 1, 1, . . .
Diagonal 2:
1, 2, 3, 4, 5, . . . ...
Diagonal 3: 1, 3, 6, 10, 15, 21, . . .
Diagonal 4: 1, 4, 10, 20, 35, 56, . . .
(yawn)
Natural Numbers
Triangle Numbers.
Tetrahedral Numbers.
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Triangle Numbers
T1 = 1
T2 = 1 + 2 = 3
T3 = 1 + 2 + 3 = 6
T4 = 1 + 2 + 3 + 4 = 10
The n-th triangle number is the sum of the first n natural
numbers
1
1
1
1
1
2
3
1
3
1
,
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
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Tetrahedral Numbers
Let Hn denote the n-th tetrahedral number. H1 = 1, H2 = 4,
H3 = 10,...
There are two ways to define Hn .
Algebraic: The n-th tetrahedral number Hn is the sum of the
first n triangle numbers.
Geometric: The n-th tetrahedral number Hn is the number
of balls required to form a tetrahedral stack with n balls
along each edge.
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Fourth Tetrahedral Number
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Sum of the Diagonal Numbers in Pascal’s Triangle
Natural Numbers: 1,2,3,4,5,...
Triangle Numbers: 1,3,6,10,15,...
Tetrahedral Numbers: 1,4,10,20,35,...
• A Triangle Number is obtained by adding Natural numbers.
• A Tetrahedral Number is obtained by adding Triangle
numbers.
What comes next?
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Sum of the Diagonal Numbers in Pascal’s Triangle
Natural Numbers: 1,2,3,4,5,...
Triangle Numbers: 1,3,6,10,15,...
Tetrahedral Numbers: 1,4,10,20,35,...
• A Triangle Number is obtained by adding Natural numbers.
• A Tetrahedral Number is obtained by adding Triangle
numbers.
What comes next?
• A Pentatopic Number is obtained by adding Tetrahedral
numbers.
1+4=5
1 + 4 + 10 = 15
1 + 4 + 10 + 20 = 35
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The Hockey Stick Identity:
Rule: The sum of the numbers along a diagonal of Pascal’s
Triangle equals the number in the next row shifted one
place to the right (for Southwesterly diagonals).
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
1 8 28 56 70 56 28 8 1
Example: 1 + 4 + 10 + 20 = 35
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Binomial Expansions:
(x
(x
(x
(x
(x
+ y )0
+ y )1
+ y )2
+ y )3
+ y )4
=
1
=
1x + 1y
=
1x 2 + 2xy + 1y 2
=
1x 3 + 3x 2 y + 3xy 2 + 1y 3
4
= 1x + 4x 3 y + 6x 2 y 2 + 4xy 3 + 1y 4
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Binomial Expansions:
(x
(x
(x
(x
(x
+ y )0
+ y )1
+ y )2
+ y )3
+ y )4
=
1
=
1x + 1y
=
1x 2 + 2xy + 1y 2
=
1x 3 + 3x 2 y + 3xy 2 + 1y 3
4
= 1x + 4x 3 y + 6x 2 y 2 + 4xy 3 + 1y 4
Rule: The coefficients in the binomial expansion of (x + y )n
are the numbers in the n-th row of Pascal’s Triangle.
Lets see if we can understand this.
(x + y )2 = (x + y )(x + y )
= x 2 + yx + xy + y 2 ,
= x 2 + 2xy + y 2 ,
FOIL, Distributive Law
Commutative law
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Calculating (x + y )3 and (x + y )4
Lets calculate (x + y )3 , given (x + y )2 :
(x + y )3 = (x + y )2 (x + y ) = (x 2 + 2xy + y 2 )(x + y )
= (x 2 + 2xy + y 2 )x + (x 2 + 2xy + y 2 )y
Once more: Lets calculate (x + y )4 , given (x + y )3 :
(x + y )4 = (x + y )3 (x + y ) = (x 3 + 3x 2 y + 3xy 2 + y 3 )(x + y )
= (x 3 + 3x 2 y + 3xy 2 + y 3 )x + (x 3 + 3x 2 y + 3xy 2 + y 3 )y
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Using Pascal’s Triangle to calculate (x + y )6
Example: Use Pascals Triangle to find (x + y )6 .
1
1
1
1
1
2
3
1
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
1 8 28 56 70 56 28 8 1
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The Magic Number Eleven
Calculate 11, 112 , 113 , and find the pattern.
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The Magic Number Eleven
Calculate 11, 112 , 113 , and find the pattern.
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Sum of the Numbers in a Row of Pascal’s Triangle.
1
1
1
1
1
2
3
1
3
1
,
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Row
R0
R1
R2
R3
R4
R5
R6
..
.
Sum
1
1+1
1+2+1
Total
1
2
4
Rule: The sum of the numbers in the n-th row of Pascal’s
Triangle is
WHY?
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The Shallow Diagonals of Pascal’s Triangle
A shallow diagonal is formed by moving left one unit and down
a diagonal one unit to get to the next number.
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
1 8 28 56 70 56 28 8 1
Sum
1
1+1
2+1
1+3+1
Total
1
2
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The Shallow Diagonals of Pascal’s Triangle
A shallow diagonal is formed by moving left one unit and down
a diagonal one unit to get to the next number.
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
1 8 28 56 70 56 28 8 1
Sum
1
1+1
2+1
1+3+1
Total
1
2
Rule: The sums of the numbers on the shallow diagonals are
the
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Sum of squares of numbers in a row.
1
1
1
1
1
2
3
1
3
1
,
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Sum of Squares
12
12 + 12
12 + 22 + 12
12 + 32 + 32 + 12
12 + 42 + 62 + 42 + 12
Total
=1
=2
=6
= 20
= 70
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Sum of squares of numbers in a row.
1
1
1
1
1
2
3
1
3
1
,
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Sum of Squares
12
12 + 12
12 + 22 + 12
12 + 32 + 32 + 12
12 + 42 + 62 + 42 + 12
Total
=1
=2
=6
= 20
= 70
Rule: The sum of the squares of the numbers in n-th row Rn is
the number in the middle of row R2n .
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Colored Pascal’s Triangle
http://oldweb.cecm.sfu.ca/cgi-bin/organics/pascalform
I: Two-colored Pascal Triangle:
Paint any even number black, any odd number red
Why do we get lots of big black triangles appearing ?
II: Three-colored Pascal Triangle:
Black: 3,6,9,12,15,18,...
Red: 1,4,7,10,13,16,19,...
Green: 2,5,8,11,14,17,20,...
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How many odd numbers in a given row?
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
1 8 28 56 70 56 28 8 1
row #
0
1
2
3
4
5
6
7
row # in binary
#odd
1
2
2
4
2
4
4
8
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How many odd numbers in a given row?
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
1 8 28 56 70 56 28 8 1
row #
0
1
2
3
4
5
6
7
row # in binary
#odd
1
2
2
4
2
4
4
8
The number is always a power of 2.
Binary expansion: 5 = 1 · 4 + 0 · 2 + 1 · 1 = 101two .
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Prime Numbered rows:
What property do all the numbers in a prime numbered row
have? Ignore the 1’s at the two ends.
Second row: 1,2,1
Third row: 1,3,3,1
Fifth row: 1,5,10,10,5,1
Seventh row: 1,7,21,35,35,21,7,1
Eleventh row: 1,11,55,165,330,462,462,330,165,55,11,1
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Prime Numbered rows:
What property do all the numbers in a prime numbered row
have? Ignore the 1’s at the two ends.
Second row: 1,2,1
Third row: 1,3,3,1
Fifth row: 1,5,10,10,5,1
Seventh row: 1,7,21,35,35,21,7,1
Eleventh row: 1,11,55,165,330,462,462,330,165,55,11,1
If p is a prime then every number in row p (except the 1’s) is
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