# A3a - n columns, exactly l of which are empty, and...

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DUE FRIDAY, 22 OCTOBER AT 10:31PM (1) (15 points) Let K n denote the complete graph on n vertices. (a) (7 points) Show that the number of ways of covering K n with paths of vertex-length 1 or more is ± x n n ! ² exp ³ x (2 - x ) 2 (1 - x ) ´ . (b) (8 points) Show that the number of ways of covering K n with cycles of length 3 or more is ± x n n ! ² (1 - x ) - 1 / 2 exp ³ - x 2 - x 2 4 ´ (2) (15 points) A simple graph is a graph with no loops or multiple edges. (a) (7 points) Show that the number of simple connected labelled graphs on n vertices and i edges is ± y i x n n ! ² log X m 0 x m m ! (1 + y ) ( m 2 ) . (b) (8 points) Show that the number of simpled labelled graphs with k components on n vertices and i edges is ± y i x n n ! ² 1 k ! log X m 0 x m m ! (1 + y ) ( m 2 ) k (3) (15 points) Show that the number of { 0 , 1 } -matrices with m rows, exactly k of which are empty (contain only 0s),

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Unformatted text preview: n columns, exactly l of which are empty, and containing p 1s is (-1) m + n + k + l X i,j (-1) i + j m i n j m-i k n-j l ij p . (4) (15 points) Consider n lines in general position, so that no three are concurrent. A frame consists of n of the ( n 2 ) points of intersection, such that no three of the n points lie on the same line. Show that the number of frames on n lines is x n n ! (1-x )-1 / 2 exp -x 2-x 2 2 . 1 2 DUE FRIDAY, 22 OCTOBER AT 10:31PM (5) (15 points) Show that the number of sequences on { 1 , . . . , n } of length m and containing k dierent objects is n k x m m ! ( e x-1) k ....
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## This note was uploaded on 11/18/2010 for the course CO 330 taught by Professor R.metzger during the Spring '05 term at Waterloo.

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A3a - n columns, exactly l of which are empty, and...

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