## Question

(a) How many ways can I select and rank 20 books to be my favorites from my list of 50? For instance: Moby Dick is #1, Tom Sawyer is #2, Brave New World is #3, etc.

(b) I can’t decide whether I prefer Tom Sawyer, Huckleberry Finn, or A Connecticut Yankee in King Arthurs Court, so I rate them as a 3-way tie. How many ways can I select and rank 20 books to be my favorites if these three are in the top 20, and these are the only ties?

(c) Once I’ve come up with a ranking of top 20 I then turn to the remaining 30 books. I group them in three categories, rather than ranking them precisely: #21 - 30, #31 - 40, #41 - 50. How many ways can this be done? As in all questions on the exam, you must justify your answer (show work) to get credit.

(d) If my friend randomly chooses a book to read from my top 50 then what is the chance it’s among my 20 favorites?

(e) If my friend randomly chooses two different books to read from my top 50 then what is the chance that neither are among my 20 favorites?

2. Suppose there are 10 men and 9 women in this class. I put your names in a hat, pick a name to give extra credit to, return the name to the hat (so I might pick it again), choose a new random name, and repeat some number of times.

(a) If I choose two times then how many ways can I choose two men or two women? What is the probability that this happens?

Note: You will have two answers. One is a single number, as in “There are 10*3—9*2 = 12 ways to choose two men or two women.” The other is a probability, as in “The probability is 12/120 = 10%.”

(b) Suppose that Alice and Bob’s names were on unusually large pieces of paper, so that each time there is a 10% chance I pick Alice, a 10% chance I pick Bob, and a 4.7% chance for each of the other 17 students. If I choose three times then what is the chance that I choose Alice once and you once? Note: So the other time a person other than Alice or you will be chosen.

3. What is the coefficient of z³ in the expansion of (3z — 2)⁷? You must justify your answer, because if you write 17 z? then for all I know you just typed it into a software package and don’t know why it’s true.

4. Suppose that Poker chips come in 5 colors: Red. Blue, Green. White. and Black. A young child wants to make pretty patterns by lining them up, but doesn’t like White following White, or Black following Black. Write a recurrence relation P, for the number of ways the child can make a line including n poker chips. without consecutive White or Black chips.

Note: Assume that the child has as many chips as they want in every color, so the only limitation is that they don’t like White-White or Black-Black.

(a) Set up a recurr³ip sequences satisfy the child’s conditions? Again, show work.

(ii) How many 20 chips sequences are possible (without the child’s conditions)? Again. show work.

(iii) How can you find the probability that a random sequence satisfies the child’s conditions?

(d) Suppose (incorrectly) that the recurrence was Pₙ = 6 Pₙ₋₁ — 8 Pₙ₋₂ with P₀ = 3 and P₁ = 8. Use methods we learned to solve this recurrence relation. Again. show work.

Extra:

Using g Rules of Products/Sums. Do the following:

1. Give a counting problem that interests you, and find the solution (with explanation). Your problem should differ somewhat from examples I did or that are in the book, though it can be a minor difference such as "Roll two dice and have total of 10" instead of "and have total of 7." Common examples are lotteries, rolling dice, or odds of interesting hands of cards (eg. Full House).

2. Now do the same, but with uneven probabilities. For instance, "Boys born with probability 0.51, girls with 0.49, find the probability that if a family has 3 kids then two will be boys and one a girl.

3. Find the product AB.

A = [ 0, -1 & 7, 2 & -4, -3 ]

B = [4, -1, 2, 3, 0 & -2, 0, 3, 4, 1]

4. Let A = [-1, 2 & 1, 3]. Find A³.

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