184a_99f_1e

184a_99f_1e - Math 184A First Exam 27 October 1999 1...

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Math 184A First Exam 27 October 1999 1. Calculate the number of 6 card hands that contain (a) two 3-of-a-kind; e.g., 3 eights and 3 jacks; (b) two pair, but not three pair or 3 of a kind. 2. Give an example of each of the following, or explain why no such example can exist. (a) An ordered list of length 5 with no repeats chosen from a set of 4 elements. (b) A 5 vertex, 5 edge simple graph that is not connected. (c) A 4 vertex, 10 edge simple graph that is connected. 3. Recall that a simple graph with vertices V is ( V, E ) where the edges E are a set chosen from P 2 ( V ) and a simple directed graph ( ) has edges chosen form V × V . (a) A simple multigraph with vertices V is ( ) where the edges E are a multiset chosen from P 2 ( V ). Prove that the number of simple multigraphs with vertices V = { 1 ,...,n } and q edges is ( N + q - 1 q ) where N = ( n 2 ) . (b) A simple directed multigraph with vertices V is ( ) where the edges E are a multiset chosen from V × V . Find and prove a formula for the number of simple directed multigraphs with vertices V = { 1 , 2 } and q edges. 4. A partition of a positive integer is an unordered list of positive integers whose sum is
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