184a_07w_fe

# 184a_07w_fe - Math 184A Final Exam(100 points 21 March 2007...

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Math 184A Final Exam (100 points) 21 March 2007 Please put your name and ID number on your blue book. CLOSED BOOK except for BOTH SIDES of one page of notes. Calculators are NOT allowed. You must show your work to receive credit. 1. (18 pts.) Let S ( n,k ) be the Stirling numbers of the second kind — the number of partitions of an n -set into k blocks. Derive the following (a) S ( n, 2) = 2 n - 1 - 1 (b) S ( n,n - 1) = n ( n - 1) 2 . 2. (10 pts.) A long rectangular table has n seats on each side and none at the ends. We want to seat n men and n women at the table so that no one is next to or opposite a person of the same sex. How many ways can this be done? Note : There are no symmetries—the sides of the table are diﬀerent. 3. (14 pts.) Each edge of a simple graph G is assigned a weight from the set { 1 , 2 ,..., 9 } and each of the nine weights is used at least once. (a) If the graph G has 10 (ten) edges, what is the largest number of minimum weight spanning trees it can have? Why?

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184a_07w_fe - Math 184A Final Exam(100 points 21 March 2007...

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