binomial coefficients and combinatorial...
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![Page 1: Binomial Coefficients and Combinatorial Identitiesirani/w17-6D/BoardNotes/23_BinomialCoefficientsPost.pdf · To generalize…. •(x + y)n = sum over 2n terms, each of which is a](https://reader030.vdocument.in/reader030/viewer/2022040204/5ea9a33e70c99527df1d3c5a/html5/thumbnails/1.jpg)
Binomial Coefficients and Combinatorial Identities
ICS 6D
Sandy Irani
![Page 2: Binomial Coefficients and Combinatorial Identitiesirani/w17-6D/BoardNotes/23_BinomialCoefficientsPost.pdf · To generalize…. •(x + y)n = sum over 2n terms, each of which is a](https://reader030.vdocument.in/reader030/viewer/2022040204/5ea9a33e70c99527df1d3c5a/html5/thumbnails/2.jpg)
• Multiply the following polynomial:
(x + y)(x + y) =
(x + y)3 = (x + y)(x + y)2
![Page 3: Binomial Coefficients and Combinatorial Identitiesirani/w17-6D/BoardNotes/23_BinomialCoefficientsPost.pdf · To generalize…. •(x + y)n = sum over 2n terms, each of which is a](https://reader030.vdocument.in/reader030/viewer/2022040204/5ea9a33e70c99527df1d3c5a/html5/thumbnails/3.jpg)
(x + y)3 = x3 + 3x2y + 3xy2 + y3
To get the coefficient of the x2y term:
xxy
xyx
yxx
![Page 4: Binomial Coefficients and Combinatorial Identitiesirani/w17-6D/BoardNotes/23_BinomialCoefficientsPost.pdf · To generalize…. •(x + y)n = sum over 2n terms, each of which is a](https://reader030.vdocument.in/reader030/viewer/2022040204/5ea9a33e70c99527df1d3c5a/html5/thumbnails/4.jpg)
To generalize….
• (x + y)n = sum over 2n terms, each of which is
a “string” of length n over {x, y}
Coefficient of xkyn-k = the number of strings of length n with k x’s and n-k y’s
![Page 5: Binomial Coefficients and Combinatorial Identitiesirani/w17-6D/BoardNotes/23_BinomialCoefficientsPost.pdf · To generalize…. •(x + y)n = sum over 2n terms, each of which is a](https://reader030.vdocument.in/reader030/viewer/2022040204/5ea9a33e70c99527df1d3c5a/html5/thumbnails/5.jpg)
The Binomial Theorem
• For any x and y, and any natural number n
(𝑥 + 𝑦)𝑛 = 𝑛𝑘𝑥𝑘𝑦𝑛−𝑘
𝑛
𝑘=0
![Page 6: Binomial Coefficients and Combinatorial Identitiesirani/w17-6D/BoardNotes/23_BinomialCoefficientsPost.pdf · To generalize…. •(x + y)n = sum over 2n terms, each of which is a](https://reader030.vdocument.in/reader030/viewer/2022040204/5ea9a33e70c99527df1d3c5a/html5/thumbnails/6.jpg)
Apply to (x + y)5
(𝑥 + 𝑦)𝑛 = 𝑛𝑘𝑥𝑘𝑦𝑛−𝑘
𝑛
𝑘=0
![Page 7: Binomial Coefficients and Combinatorial Identitiesirani/w17-6D/BoardNotes/23_BinomialCoefficientsPost.pdf · To generalize…. •(x + y)n = sum over 2n terms, each of which is a](https://reader030.vdocument.in/reader030/viewer/2022040204/5ea9a33e70c99527df1d3c5a/html5/thumbnails/7.jpg)
Apply to (3a – 2b)6
(𝑥 + 𝑦)𝑛 = 𝑛𝑘𝑥𝑘𝑦𝑛−𝑘
𝑛
𝑘=0
![Page 8: Binomial Coefficients and Combinatorial Identitiesirani/w17-6D/BoardNotes/23_BinomialCoefficientsPost.pdf · To generalize…. •(x + y)n = sum over 2n terms, each of which is a](https://reader030.vdocument.in/reader030/viewer/2022040204/5ea9a33e70c99527df1d3c5a/html5/thumbnails/8.jpg)
Apply to (-4a + 3b)9
(𝑥 + 𝑦)𝑛 = 𝑛𝑘𝑥𝑘𝑦𝑛−𝑘
𝑛
𝑘=0
*
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Binomial Theorem for Identities
Plug in x = y = 1
(𝑥 + 𝑦)𝑛 = 𝑛𝑘𝑥𝑘𝑦𝑛−𝑘
𝑛
𝑘=0
*
![Page 10: Binomial Coefficients and Combinatorial Identitiesirani/w17-6D/BoardNotes/23_BinomialCoefficientsPost.pdf · To generalize…. •(x + y)n = sum over 2n terms, each of which is a](https://reader030.vdocument.in/reader030/viewer/2022040204/5ea9a33e70c99527df1d3c5a/html5/thumbnails/10.jpg)
Combinatorial Argument for Identities
2𝑛 = 𝑛𝑘
𝑛
𝑘=0
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 𝑜𝑓 1, 2, 3,… , 𝑛
= 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑘 − 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 𝑜𝑓 1, 2, 3, . . , 𝑛
𝑛
𝑘=0
![Page 11: Binomial Coefficients and Combinatorial Identitiesirani/w17-6D/BoardNotes/23_BinomialCoefficientsPost.pdf · To generalize…. •(x + y)n = sum over 2n terms, each of which is a](https://reader030.vdocument.in/reader030/viewer/2022040204/5ea9a33e70c99527df1d3c5a/html5/thumbnails/11.jpg)
Pascal’s Identity
# k-subsets of {1, 2, …,n, n+1}
# k-subsets of {1, 2, …,n, n+1}
that do not include 1
# k-subsets of {1, 2, …,n, n+1}
that DO include 1
= +
Example: n = 4, k = 3 3-subsets from {1, 2, 3, 4, 5}
{1, 2, 3} {1, 2, 4} {1, 2, 5} {1, 3, 4} {1, 3, 5} {1, 4, 5}
{2, 3, 4} {2, 3, 5} {2, 4, 5} {3, 4, 5}
![Page 12: Binomial Coefficients and Combinatorial Identitiesirani/w17-6D/BoardNotes/23_BinomialCoefficientsPost.pdf · To generalize…. •(x + y)n = sum over 2n terms, each of which is a](https://reader030.vdocument.in/reader030/viewer/2022040204/5ea9a33e70c99527df1d3c5a/html5/thumbnails/12.jpg)
Pascal’s Identity
# k-subsets of {1, 2, …,n, n+1}
# k-subsets of {1, 2, …,n, n+1}
that do not include 1
# k-subsets of {1, 2, …,n, n+1}
that DO include 1
= +
![Page 13: Binomial Coefficients and Combinatorial Identitiesirani/w17-6D/BoardNotes/23_BinomialCoefficientsPost.pdf · To generalize…. •(x + y)n = sum over 2n terms, each of which is a](https://reader030.vdocument.in/reader030/viewer/2022040204/5ea9a33e70c99527df1d3c5a/html5/thumbnails/13.jpg)
Pascal’s Triangle