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184a_05f_fe

# 184a_05f_fe - Math 184A Final Exam(80 points 8AM 7 Dec 2005...

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Math 184A Final Exam (80 points) 8AM 7 Dec. 2005 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. (16 pts.) A square table has two seats on each side for a total of eight seats. Rotations of the table don’t matter. Thus, if 1 , 2 , . . ., 8 are placed around the table, 1 2 8 3 7 4 6 5 and 7 8 6 1 5 2 4 3 are the same, but di±er from 2 1 3 8 4 7 5 6 and 8 1 7 2 6 3 5 4 . (a) How many ways can eight people be seated at the table? (b) We have four identical red chairs and four identical blue chairs. How many ways can the eight chairs be placed around the table? Again, rotations of the table do not matter. 2. (18 pts.) Let V = { 1 , 2 , . . ., n } . (a) Compute the number of simple graphs with vertex set V that have exactly q edges. (b) A vertex is isolated if it does not lie on any edges. Suppose S V . Compute the number of simple graphs with vertex set

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184a_05f_fe - Math 184A Final Exam(80 points 8AM 7 Dec 2005...

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