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What is the value of a1 for the recurrence relation in item 8? (refer to choices in item 7?




How many substrings can be formed from the string SOFTWARES contain the string SOFT together in any order?




How many strings of length 4 can be formed using the letters ABCDE if it starts with letters AC and repetition is not



allowed?



2. There are 10 multiple choice questions in an examination. Each of the questions have four choices. In how many ways



can an examinee give possible answers?



B. BINOMIAL COEFFICIENTS



Expand 2� + 4�



4 using the binomial theorem. (10 pts)



C. PIGEONHOLE PRINCIPLE (5 pts) . Explain briefly. Do you agree that there are 3 persons who have the same first and last name? Why and why not?

At the beginning of the first day (day 1) after grape harvesting is completed, a grape grower has 8000 kg of grapes in storage. On day n, for n = 1, 2, . . . , the grape grower sells 250n/(n + 1) kg of the grapes at the local market at the price of $2.50 per kg. He leaves the rest of the grapes in storage where each day they dry out a little so that their weight decreases by 3%. Let wn be the weight (in kg) of the stored grapes at the beginning of day n for n ≥ 1 (before he takes any to the market).




(a) Find the value of wn for n = 2.




(b) Find a recursive definition for wn. (You may find it helpful to draw a timeline.)




(c) Let rn be the total revenue (in dollars) earned from the stored grapes from the beginning of day 1 up to the beginning of day n for n ≥ 1. Find a recursive formula for rn.











Some time light fixtures are controlled by more than one switch.circuit's need to be designed so that flipping any one of the switches for the fixtures turn the light on when it is off and turn the light off when it is on.design circuits that accomplish this when there are two switches and when there are three switches

Find, showing all working, a recursive de nition of the sequence with general

term

tn = 3 (n + 1)!/2n ; n >= 1


a) Is function g defined by g = {(-1 , 2),(0 , 4),(2 , -4),(5 , 6),(10 , 0)}, a one to one function?


b) State whether the given function is on-to or not. f : R -> R defined by f(x) = 1 + x2.


c) Let A = {3, 4, 5} and B= {6, 7, 8, 9} and R = {(3, 6) (4, 7) (5, 8)}. Find out if R is a mapping from A to B.


d) The following sets have been defined using the | notation. Re-write them by listing some of the elements.

                   i.  {x | x = 2n - 5, x and n are natural numbers}

                  ii.  {y | 2y2 = 50, y is an integer}





What is the difference between an outcome and an event?


show that p↔q and (p∧q)∨(¬p∧¬q) are logically equivalent


Show that ¬(¬p) and p are logically equivalent.


Choose the best answer: ANTI SYMMETRIC, TRANSITIVE RELATION, SYMMETRIC RELATION, REFLEXIVE RELATION



1. {(3,5)(5,3) (2,4)(4,5)}



2. {(3,1)(1,3)}



3. {(3,1)(2,3)(5,6)(6,5)}



4. {(1,3)(5,3)(7,5)}



5. {(8,9)(9,7)(8,7)}



6. {(4,2)(2,4)}



7. {(1,5)(2,5)(3,5)}



8. {(5,5)(5,6)(6,5)(6,6)(6,7)(7,6) (7,7)}



9. {(6,5)(5,4)(6,4)}



10. {(7,6)(6,5)(7,5)}



11. {(3,1)(1,3)}



12. {(4,3)(3,5)(4,5)}



13. {(6,6)(6,5)(5,5)(5,4)(4,4)}



14. {(6,5)(7,6)(4,5)(5,4)}



15. {(3,3)(4,4)(4,5)(5,4)(5,5)}


A survey of 100 college students revealed the following information about their enrolment in Mathematics, Political Science, and Sociology courses. Twenty-six take Mathematics; 65 take Political Science; 65 take Sociology; 14 take Mathematics and Political Science; 13 take Mathematics and Sociology; 40 take Political Science and Sociology; 8 take Mathematics, Political Science and Sociology. How many students are taking both Mathematics and Sociology but not Political Science?


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