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Pascal's Triangle
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Section 8-7Pascal’s Triangle
Warm-upGive the number of subsets of the set of letters {A, E, I, O, U, and Y} that have the given number of
elements.
1. o elements 2. 1 element 3. 2 elements
4. 3 elements 5. 4 elements 6. 5 elements
7. 6 elements
Warm-upGive the number of subsets of the set of letters {A, E, I, O, U, and Y} that have the given number of
elements.
1. o elements 2. 1 element 3. 2 elements
4. 3 elements 5. 4 elements 6. 5 elements
7. 6 elements
1
Warm-upGive the number of subsets of the set of letters {A, E, I, O, U, and Y} that have the given number of
elements.
1. o elements 2. 1 element 3. 2 elements
4. 3 elements 5. 4 elements 6. 5 elements
7. 6 elements
1 6
Warm-upGive the number of subsets of the set of letters {A, E, I, O, U, and Y} that have the given number of
elements.
1. o elements 2. 1 element 3. 2 elements
4. 3 elements 5. 4 elements 6. 5 elements
7. 6 elements
1 6 15
Warm-upGive the number of subsets of the set of letters {A, E, I, O, U, and Y} that have the given number of
elements.
1. o elements 2. 1 element 3. 2 elements
4. 3 elements 5. 4 elements 6. 5 elements
7. 6 elements
1 6 15
20
Warm-upGive the number of subsets of the set of letters {A, E, I, O, U, and Y} that have the given number of
elements.
1. o elements 2. 1 element 3. 2 elements
4. 3 elements 5. 4 elements 6. 5 elements
7. 6 elements
1 6 15
20 15
Warm-upGive the number of subsets of the set of letters {A, E, I, O, U, and Y} that have the given number of
elements.
1. o elements 2. 1 element 3. 2 elements
4. 3 elements 5. 4 elements 6. 5 elements
7. 6 elements
1 6 15
20 15 6
Warm-upGive the number of subsets of the set of letters {A, E, I, O, U, and Y} that have the given number of
elements.
1. o elements 2. 1 element 3. 2 elements
4. 3 elements 5. 4 elements 6. 5 elements
7. 6 elements
1 6 15
20 15 6
1
Warm-upGive the number of subsets of the set of letters {A, E, I, O, U, and Y} that have the given number of
elements.
1. o elements 2. 1 element 3. 2 elements
4. 3 elements 5. 4 elements 6. 5 elements
7. 6 elements
1 6 15
20 15 6
11, 6, 15, 20, 15, 6, 1
Pascal’s Triangle
Pascal’s Triangle
The pattern that emerges from displaying rows of combinations
Pascal’s Triangle
The pattern that emerges from displaying rows of combinations
Let’s build it!
Interactive
1
Interactive
11
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111
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111
1
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111
1 2
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111
1 2 1
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111
1 2 11
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111
1 2 11 3
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111
1 2 11 3 3
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111
1 2 11 3 3 1
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111
1 2 11 3 3 11 4 6 4 1
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111
1 2 11 3 3 11 4 6 4 11 5 10 10 5 1
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111
1 2 11 3 3 11 4 6 4 11 5 10 10 5 1
1 6 15 20 15 6 1Interactive
(r + 1)st term in row n in Pascal’s Triangle
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(r + 1)st term in row n in Pascal’s Triangle
To find a particular term in Pascal’s Triangle, let n be the row. Then the value of the (r + 1)st term is nCr.
Interactive
(r + 1)st term in row n in Pascal’s Triangle
To find a particular term in Pascal’s Triangle, let n be the row. Then the value of the (r + 1)st term is nCr.
Check it out if you don’t believe me.
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Properties of Pascal’s Triangle
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Properties of Pascal’s Triangle1. The first and last terms of each row are 1’s
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Properties of Pascal’s Triangle1. The first and last terms of each row are 1’s
2. The second and next-to-last terms in the nth row are n
Interactive
Properties of Pascal’s Triangle1. The first and last terms of each row are 1’s
2. The second and next-to-last terms in the nth row are n
3. The rows are symmetrical
Interactive
Properties of Pascal’s Triangle1. The first and last terms of each row are 1’s
2. The second and next-to-last terms in the nth row are n
3. The rows are symmetrical
4. The sum of the terms in row n is 2n
Interactive
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term:
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term: 9C0
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term: 9C0 =
9!9 − 0( )!0!
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term: 9C0 =
9!9 − 0( )!0!
=9!9!
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term: 9C0 =
9!9 − 0( )!0!
=9!9! = 1
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term: 9C0 =
9!9 − 0( )!0!
=9!9! = 1
Second term:
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term: 9C0 =
9!9 − 0( )!0!
=9!9! = 1
Second term: 9C1
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term: 9C0 =
9!9 − 0( )!0!
=9!9! = 1
Second term: 9C1 = 9
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term: 9C0 =
9!9 − 0( )!0!
=9!9! = 1
Second term: 9C1 = 9
Third term:
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term: 9C0 =
9!9 − 0( )!0!
=9!9! = 1
Second term: 9C1 = 9
Third term: 9C2
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term: 9C0 =
9!9 − 0( )!0!
=9!9! = 1
Second term: 9C1 = 9
Third term: 9C2 = 36
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term: 9C0 =
9!9 − 0( )!0!
=9!9! = 1
Second term: 9C1 = 9
Third term: 9C2 = 36
Fourth term:
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term: 9C0 =
9!9 − 0( )!0!
=9!9! = 1
Second term: 9C1 = 9
Third term: 9C2 = 36
Fourth term: 9C3
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term: 9C0 =
9!9 − 0( )!0!
=9!9! = 1
Second term: 9C1 = 9
Third term: 9C2 = 36
Fourth term: 9C3 = 84
Example 2Give a polynomial formula for the third term in the nth
row of Pascal’s Triangle.
Interactive
Example 2Give a polynomial formula for the third term in the nth
row of Pascal’s Triangle.
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We need something to fit the pattern:
Example 2Give a polynomial formula for the third term in the nth
row of Pascal’s Triangle.
Interactive
We need something to fit the pattern:1, 3, 6, 10, 15, ...
Example 2Give a polynomial formula for the third term in the nth
row of Pascal’s Triangle.
Interactive
We need something to fit the pattern:1, 3, 6, 10, 15, ...
Differences:
Example 2Give a polynomial formula for the third term in the nth
row of Pascal’s Triangle.
Interactive
We need something to fit the pattern:1, 3, 6, 10, 15, ...
Differences: 2, 3, 4, 5, ...
Example 2Give a polynomial formula for the third term in the nth
row of Pascal’s Triangle.
Interactive
We need something to fit the pattern:1, 3, 6, 10, 15, ...
Differences: 2, 3, 4, 5, ...1, 1, 1, ...
Example 2Give a polynomial formula for the third term in the nth
row of Pascal’s Triangle.
Interactive
We need something to fit the pattern:1, 3, 6, 10, 15, ...
Differences: 2, 3, 4, 5, ...1, 1, 1, ...
This tells us we are dealing with a 2nd degree polynomial
Example 2(2, 1), (3, 3), (4, 6), (5, 10)...
Example 2(2, 1), (3, 3), (4, 6), (5, 10)...
Example 2(2, 1), (3, 3), (4, 6), (5, 10)...
Example 2(2, 1), (3, 3), (4, 6), (5, 10)...
Example 2(2, 1), (3, 3), (4, 6), (5, 10)...
y =12 x2 − 1
2 x
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Row 12:
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Row 12:1
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Row 12:1 12
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Row 12:1 12 66
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Row 12:1 12 66 220
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Row 12:1 12 66 220 495
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Row 12:1 12 66 220 495 792
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Row 12:1 12 66 220 495 792 924
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Row 12:1 12 66 220 495 792 924 792
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Row 12:1 12 66 220 495 792 924 792 495
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Row 12:1 12 66 220 495 792 924 792 495 220
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Row 12:1 12 66 220 495 792 924 792 495 220 66
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Row 12:1 12 66 220 495 792 924 792 495 220 66 12
Example 3Here are the first 6 terms of row 11 of Pascal’s Triangle:
1, 11, 55, 165, 330, 462.Use them to construct row 12.
There are 12 terms in row 11, so the entire row is:1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Row 12:1 12 66 220 495 792 924 792 495 220 66 12 1
Homework
Homework
p. 532 #1-20
