midterm - , S 2 , S 3 , ... Question #7. 1 (a) Solve the...

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March 4, 2004 Discrete Mathematics, Spring 2004 Midterm Question #1. Find a simple theta notation for the following: (a) f ( n ) = C ( n, 100) (b) g ( n ) = 2 lg 2 + 3 lg 3 + ... + n lg n Question #2. De±ne the sequence of harmonic numbers by H 1 = 1 , H 2 = 1 + 1 2 , H 3 = 1 + 1 2 + 1 3 , · · · Show that n j =1 H j = ( n + 1) H n n for all n 1 . Question #3. In how many ways can one be dealt a 5-card poker hand (from a 52-card deck) that does not contain a pair, two pair, three of a kind, ²ush, full house, or four of a kind? (For the purposes of this question, treat a ²ush as a hand containing 5 cards of the same suit.) Question #4. Determine the number of nonnegative integer solutions to the pair of equa- tions x 1 + x 2 + ... + x 7 = 37 , x 1 + x 2 + x 3 = 6 . Question #5. Suppose n is even. Show that n/ 2 k =0 C ( n, 2 k ) = 2 n - 1 = n/ 2 k =1 C ( n, 2 k 1) . Question #6. Let S n denote the number of subsets of { 1 ,... ,n } that do not contain two consecutive numbers. Determine a recurrence relation and initial conditions for the sequence S 1
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Unformatted text preview: , S 2 , S 3 , ... Question #7. 1 (a) Solve the recurrence relation a n = a n-1 + 1 + 2 n-1 subject to the initial condition a = 1. (b) Solve the recurrence relation a n = 2 a n-1 5 a n-2 subject to the initial conditions a = 4, a 1 = 4. Question #8. Consider the following useless algorithm: 1 . procedure foo ( n ) 2 . if n = 1 then 3 . return 4 . x := x + 1 5 . m := n/ 2 6 . call foo ( m ) 7 . if n is even then 8 . call foo ( m ) 9 . else 10 . x := x + 1 11 . end foo Let b n denote the number of times the statement x := x + 1 is executed. (a) Determine a recurrence relation and initial conditions for the sequence { b n } . (b) Solve the recurrence relation in case n is a power of 2. Extra Credit. Using some calculus if necessary, Fnd a simpliFed expression for n k =1 k 2 2 k . 2...
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midterm - , S 2 , S 3 , ... Question #7. 1 (a) Solve the...

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