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- Assume that a license plate must be of length 7. How many different license plates are possible (a) if it can only contain capital letters? (b) if it can only contain capital letters, and repetition of letters isn’t allowed? (c) if it can contain capital letters or the digits 0-9, repetition isn’t allowed, and the plate must begin with the letter “R”?In how ways can the letters of the word 'ORANGE' be arranged so that theconsonants occupy only the even positions?Prove by Contrapositive: "For all integers a and b, if a+b is odd then a is odd or b is odd." (Your answer should be handwritten(readable) in any size of bond paper. Take a picture of your answer and attach it here. Digitally created answer will not be accepted).
- 17≡ 7 (mod 2) ,true or false?4 Assume that X={A,F,G,D,C} and Y={6,1,4,2,3}. A code consists of 4 different symbols selected from X followed by 2 not necessarily different symbols from Y. How many different codes are possible?6. Suppose you need to enter a 4 digit code to enter a room.(a) How many such codes exists if there are no digits appear twice in your code?(b) How many such codes have exactly 1 digit which appears twice?(c) Now suppose you need to press 4 different digits simultaneously in order to unlock the door. Does theorder matter? How many such codes exist?
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- Assume that X={G,E,D,A}X={G,E,D,A} and Y={5,6,1,2,7}Y={5,6,1,2,7}. A code consists of 2 different symbols selected from XX followed by 4 not necessarily different symbols from YY. How many different codes are possible?Assume that A={A,B,C} and B={5,4,1,9,6} A code consists of 2 different symbols selected from X followed by 2 not necessarily different symbols from Y How many different codes are possible?Suppose that n students take a quiz and their scores sum to 150. What is the smallest number n to gurantee that at least two students got the same grade (assume that the quiz score of each student is an integer between 0 and 100)? Hint: Suppose the scores are all different and use Pigeonhole Principle.