ma017-2010-hw5

ma017-2010-hw5 - MA017 2010 HOMEWORK 5 Due 2010-11-02 Tue....

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

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

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

Unformatted text preview: MA017 2010 HOMEWORK 5 Due 2010-11-02 Tue. (1) One can prove ( n k ) + ( n k +1 ) = ( n +1 k +1 ) by “pure thought” as follows. The ( k + 1)-element subsets of { 1 ,...,n +1 } are of two mutually exclusive types: those that contain n +1, and those that do not; the former are in bijection with the k-element subsets of { 1 ,...,n } , the latter with the ( k + 1)-element subsets of { 1 ,...,n } . Give similar proofs for whichever of the following combinatorial identities strike your fancy. Assume, in all formulae where ( n k ) appears, that k ≤ n . (a) ( n k ) = ( n n- k ) (b) ∑ n k =0 ( n k ) = 2 n (c) ∑ n k =0 k ( n k ) = n 2 n- 1 (d) ∑ n k =0 k 2 ( n k ) = n 2 n- 1 + n ( n- 1)2 n- 2 (e) ∑ n k =0 k 3 ( n k ) = n 2 n- 1 + 3 n ( n- 1)2 n- 2 + n ( n- 1)( n- 2)2 n- 3 (f) ∑ k ( n 2 k ) = ∑ k ( n 2 k +1 ) (g) ( 2 n 2 ) = 2 ( n 2 ) + n 2 (h) ( 2 n +2 n +1 ) = ( 2 n n +1 ) + 2 ( 2 n n ) + ( 2 n n- 1 ) (i) ∑ n k =0 (- 1) k ( n k ) = 0 (j) ( n k ) = n k ( n- 1 k- 1 ) (k) ∑ n i...
View Full Document

This document was uploaded on 03/03/2011.

Page1 / 2

ma017-2010-hw5 - MA017 2010 HOMEWORK 5 Due 2010-11-02 Tue....

This preview shows document pages 1 - 2. Sign up to view the full document.

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