week2 - S Tanny Subsets of a Set[n 1 How many k-subsets...

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
S. Tanny MAT 344F Fall, 1997 Subsets of a Set [n] 1) How many k-subsets of [n] are there? k 0, integer n = 4 12 13 14 k = 2 34 23 24 let x be the # of k-subsets Each such subset can be arranged in k! ways. Thus, x k! counts the number of ordered k-subsets of [n], which is just n k x k! = n k x = k n k! n k What is n 0 ? 0 0 ? 3 4 ? Notice: n + 1 k = n k + n k - 1 (This is called the triangle formula for binomial coefficients.) Fix your eye on the element. ( n + 1): (n + 1) is in or out of any subset n k counts all k-subsets where (n + 1) is OUT (because these are just k-subsets of [n]). n k - 1 counts all k-subsets where (n + 1) is in. By the SUM rule, this counts all k-subsets of [n + 1].
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
S. Tanny MAT 344F Fall, 1997 "Algebraic" Proof of the above identity: n k + n k - 1 = n! k!(n - k)! + n! (k - 1)!(n - k + 1)! = = n! (k - 1)!(n - k)! n + 1 k(n - k + 1) = (n + 1)! k!(n - k + 1)! n + 1 k Note: n k = n n - k Each choice of a k-subset leaves behind an (n - k) subset.
Image of page 2
S. Tanny MAT 344F Fall, 1997 Graphs of Binomial Coefficients f 2 (n) = n 2 f 3 (n) = n 3 g 2 (k) = 2 k g 3 (k) = 3 k
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
S. Tanny MAT 344F Fall, 1997 f 2 (n) = n 2 = n(n - 1) 2 f 3 (n) = n 3 = 1 6 n(n - 1) (n - 2) g 3 (r) = 3 r g 6 (r) = 6 r Unimodal:up/down Single or Double maximum
Image of page 4
S. Tanny MAT 344F Fall, 1997 Array of Binomial Coefficients n 0 = 1 n 0 1 1 1 1 2 1 1 3 3 1 1
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern