e510k - MAD 2104 Quiz 5 and Key June 3 2010 Prof S Hudson...

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MAD 2104 June 3, 2010 Quiz 5 and Key Prof. S. Hudson 1a) [20pts each] Suppose k integers are chosen from the first eleven positive integers. Show that if k 8, then some pair of the chosen integers has a sum equal to 12. 1b) Find the smallest value of k that makes the conclusion true. Justify your answer briefly. 2a) [15pts each] A coin is tossed 10 times, where each toss comes up heads or tails. How many possible outcomes are there? 2b) How many are there, with at most 2 heads ? 3a) [15pts each] Let R = { (1 , 2) , (3 , 1) , (2 , 1) , (1 , 3) , (2 , 2) } be a relation on A = { 1 , 2 , 3 } . Is R symmetric ? 3b) Is it transitive ? (justify briefly) Remarks and Answers: The average among the top 20 students was about 65 / 100. Here is a rough scale for the quiz, based mainly on that average: As 75 to 100 Bs 65 to 74 Cs 55 to 64 Ds 45 to 54 1a) The pigeons are the k chosen integers. The pigeonholes are the 6 sets { 1 , 11 } , { 2 , 10 } , { 3 , 9 } , { 4 , 8 } , { 5 , 7 } and { 6 } . By the PHP (since
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