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

Info iconThis 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].
Background image of page 1

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

View Full DocumentRight 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.
Background 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
Background image of page 3

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

View Full DocumentRight 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
Background image of page 4
S. Tanny MAT 344F Fall, 1997
Background image of page 5

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

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

This note was uploaded on 03/04/2012 for the course BIO 510 taught by Professor Miller during the Fall '06 term at Carnegie Mellon.

Page1 / 10

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online