notes 8-7
DESCRIPTION
Pascal's TriangleTRANSCRIPT
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Section 8-7Pascal’s Triangle
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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
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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
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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
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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
![Page 6: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/6.jpg)
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
![Page 7: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/7.jpg)
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
![Page 8: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/8.jpg)
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
![Page 9: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/9.jpg)
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
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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
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Pascal’s Triangle
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Pascal’s Triangle
The pattern that emerges from displaying rows of combinations
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Pascal’s Triangle
The pattern that emerges from displaying rows of combinations
Let’s build it!
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Interactive
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1 2 11 3 3 11 4 6 4 11 5 10 10 5 1
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1 2 11 3 3 11 4 6 4 11 5 10 10 5 1
1 6 15 20 15 6 1Interactive
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(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
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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.
Check it out if you don’t believe me.
Interactive
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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
Interactive
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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
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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
3. The rows are symmetrical
Interactive
![Page 35: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/35.jpg)
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
![Page 36: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/36.jpg)
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
![Page 37: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/37.jpg)
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term:
![Page 38: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/38.jpg)
Example 1Use the formula for nCr to find the first four terms in row
9 of Pascal’s Triangle.
First term: 9C0
![Page 39: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/39.jpg)
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!
![Page 40: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/40.jpg)
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!
![Page 41: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/41.jpg)
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
![Page 42: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/42.jpg)
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:
![Page 43: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/43.jpg)
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
![Page 44: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/44.jpg)
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
![Page 45: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/45.jpg)
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:
![Page 46: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/46.jpg)
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
![Page 47: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/47.jpg)
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
![Page 48: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/48.jpg)
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:
![Page 49: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/49.jpg)
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
![Page 50: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/50.jpg)
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
![Page 51: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/51.jpg)
Example 2Give a polynomial formula for the third term in the nth
row of Pascal’s Triangle.
Interactive
![Page 52: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/52.jpg)
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:
![Page 53: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/53.jpg)
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, ...
![Page 54: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/54.jpg)
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:
![Page 55: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/55.jpg)
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, ...
![Page 56: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/56.jpg)
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, ...
![Page 57: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/57.jpg)
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
![Page 58: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/58.jpg)
Example 2(2, 1), (3, 3), (4, 6), (5, 10)...
![Page 59: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/59.jpg)
Example 2(2, 1), (3, 3), (4, 6), (5, 10)...
![Page 60: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/60.jpg)
Example 2(2, 1), (3, 3), (4, 6), (5, 10)...
![Page 61: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/61.jpg)
Example 2(2, 1), (3, 3), (4, 6), (5, 10)...
![Page 62: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/62.jpg)
Example 2(2, 1), (3, 3), (4, 6), (5, 10)...
y =12 x2 − 1
2 x
![Page 63: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/63.jpg)
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.
![Page 64: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/64.jpg)
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:
![Page 65: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/65.jpg)
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
![Page 66: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/66.jpg)
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:
![Page 67: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/67.jpg)
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
![Page 68: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/68.jpg)
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
![Page 69: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/69.jpg)
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
![Page 70: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/70.jpg)
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
![Page 71: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/71.jpg)
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
![Page 72: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/72.jpg)
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
![Page 73: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/73.jpg)
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
![Page 74: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/74.jpg)
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
![Page 75: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/75.jpg)
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
![Page 76: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/76.jpg)
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
![Page 77: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/77.jpg)
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
![Page 78: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/78.jpg)
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
![Page 79: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/79.jpg)
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
![Page 80: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/80.jpg)
Homework
![Page 81: Notes 8-7](https://reader031.vdocuments.net/reader031/viewer/2022013101/55629175d8b42a68128b4ce3/html5/thumbnails/81.jpg)
Homework
p. 532 #1-20