CSE 2353 Spring 2008 Test 2 Review

Proofs & Counting Principles

1. Prover l f both solutions of x2 +- ax +-b =0 are even integers, then a and b are both even.

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2. What is the least number of integers from {I,2, ... , 20} that must be chosen so that at least one of them is divisible by 4? [Ex 11, p 437 in Discrete Mathematical Structures: Theory and Aplications, by D.S. Malik and M.K.Sen, Thomson Course Technology. 2004.]

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esc 2353. Spring 2008 2

3. How many distinct five .. letter words can be formed [rom the letters A, N, G. r\~'{ itno letter is to be used more than once ill a word? You MUST j usri fy your answer.

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CSF; 2353, Spring 2008 3

4. Prove II1al 1 1 ' I- " . (2,., + 3) 2 _ ... - I II ~ ) / < hI~ 7 were n is an integer and n >= I .

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5. What is wrong with the fol low ing proof?

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CSE 2353. Spring :WOS _ )

6, Prove that the product or two consecutive integers is even.

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csr 2353, Spring 2008 6

7. Prove for all positive integers k> I , e > k! .

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CSE 2353. Spring 200X 7

8. If (./ is a positive Real number where a-: I , prove that j;; > ((.

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CSE 2353, Spring 2008 8

9. In a class 01'20 students. 5 will be chosen to have a picture taken. The picture will be made with the students seated in a row. How many different arrangements of the student s is possible?

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CSI: 2353. Spring 2008 9

I I. Prove that any subset of size 6 from -[ 1.2.3.4.5.6.7,8.9} must contain two elelmtnets \\ hose sum is 10.

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CSE 2353. Spring 2008 II