1. Let S-AC, Jessie, Kelly, Lisa, Screech, Zack) a. How many 3-entry (ordered) l
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1. Let S-AC, Jessie, Kelly, Lisa, Screech, Zack) a. How many 3-entry (ordered) lists of elements of S are there? ["Screech, Screech, Kelly" is a different ordered list from "Screech, Kelly, Screech".] b. How many 3-entry (ordered) lists of distinct elements of S are there? c. How many 3-element subsets of S are there? 2. How many different usernames are there if a. a username must be a single capital letter or a single digit (like "Z" or "0")? b. a username must be a single capital letter followed by a single digit (like "FO")? c. a username must be three capital letters followed by three digits (like "ABA101")? d. a username must consist of three capital letters and one digit (like "CB3C")? e. a username must consist of exactly five capital letters and exactly four digits (like "13ABC3B7Q")? Hint: first determine the letters in order and digits in order. Then shuffle them, maintaining order.] 3. Use the fact that C(n. k) (n-k) k! to show that C(n, k) = C(n. n k). [Notice that this gives another proof of the fact that the number of k-element subsets of an n-element set equals the number of (n - k)-element subsets of an n-element set. We used the Bijection Theorem in class to deduce the same result.] 4. There are 6 anagrams (rearrangements) of the word BAT. They are: ABT, ATB, BAT, BTA, TAB, TBA. (We include the word BAT itself. The anagrams need not be actual words.) How many ways can you anagram the following words? a. COMPUTER b. ANAGRAM [In class. we used a certain argument to show that C(7.3)-31 = P(7.3) Make a similar argument here. Hint: the process of anagramming 1N2GR3M can be accomplished by anagram ming ANAGRAM and then changing the As to 1.2 c. MATHEMATICS 5. How many ways are there to seat five peo ple around a circular table, where seatings are considered to be the same if they can be obtained from each other by rotating the table? 6. Each letter (A-Z) is represented in Morse code by a sequence of signals (dots or dashes). Show that each letter can be assigned its own sequence of 4 signals. You shouldn't have to look up Morse code to answer this question 7. How many different last columns occur among all the truth tables with propositional variables p, q, r, s? (In other words, how many equivalence classes are there under the relation of logical equivalence?) The answer is not 16 a. You enter a game with $100. After each round you either gain $1 or lose S1. After 10 rounds, you have $100. How many ways can this happen? [For example, one possible 8. outcome is Win,Lose, Lose,Lose, Win, Win,Lose, Win Lose, Win.]Explanation / Answer
Ans: 1) Given S= {AC, Jessie, Kelly, Lisa, Screech, Zack }
a) The number of 3 entry list of elements from the above list is 6P3= 120
b) The number of 3 entry lists wth distinct elements is 6C3= 20
c) The number of 3 elements subsets of S is 20 because the order of elements does not matter in that case.
2. a) In case when the usernames can be a single capital letter or a single digit, then there are 26 + 10 poosiblities = 36 cases.
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