Problem-solving with Combinations, Permutations, and Counting Techniques in Number Theory

Formulas to use: Combinations, Permutations, Partial Permutations, Factorial, Factorial Permutations, Counting: subsets, distinct subsets, sequences, cardinality of powerset

To do: Please justify why you are using a certain rule (i.e. multiplication rule), by correctly using its definition, Break all problems into cases (Proofs with cases: choose between proofs by-cases or If-then statement proofs)

Skills: Logical/Quantifier and English notation, Latex, Number Theory to prove problems.

TIP: Please use Latex for formulas / Save document into a word doc.

1 - Thirteen Dwarves (Thorin, Fili, Kili, Balin, Dwalin, Oin, Gloin, Dori, Nori, Ori, Bifur, Bofur, and Bombur) want to ask for Bilbo’s help, but they won’t all fit in his Hobbit-hole, so they need to choose a delegation to visit him. How many different groups with at least one member and at most twelve members can the thirteen Dwarves choose among themselves? 

2 - John has a string of n empty light bulb sockets hanging above his front door, where n = 2k is some even positive integer. He has an unlimited supply of blue, green, red, and yellow light bulbs. He wants to place the bulbs in the sockets in a symmetric way, meaning the order of colors should be the same left-to-right and right-to-left. How many different symmetric ways are there for John to arrange n bulbs of those four colors on this string?

3 - Jocelyn plans to watch all the films on a list of the 100 greatest films. She has only one constraint on the order in which she views them: Four films on the list were directed by Alfred Hitchcock, and she wants to have a “Hitchcock marathon” in which she watches those four films consecutively, in any order.

How many possible orders for the 100 movies are there that satisfy this constraint?

4 - Two boys are trick-or-treating on Halloween, dressed as T’Challa and Dracula. Mrs. Owens has a bowl with seven different candy bars in it (one Kit-Kat, one Milky Way, one Snickers, one Crunch, one Twix, one Baby Ruth, and one Almond Joy), and she tells the boys that they can each take two candy bars from the bowl. If T’Challa picks first, how many ways are there for them to select their candy?

5 - Write the following statements in English. Then say whether each statement is true or false, and briefly explain why.

(a) ∀m ∈ Z + ∃n ∈ Z (n > m ∧ n < 2m)

(b) ∀m ∈ Z ∃n ∈ Z ∀p ∈ Z p + n > m

(c) ∃m ∈ Z ∀n ∈ Z mn = m

Please don’t forget to explain why statements are true or false by using knowledge of number theory.For all positive integers m, there exist some integer n, such that mn are between n and 2m (correct me if I am wrong)

6 - Ms. Frizzle is planning to give her class a multiple-choice exam with kn problems, each with k choices, for some positive integers n and k. She wants to make sure that each of the k choices is the correct answer for exactly n of the problems. How many possible answer keys are there that satisfy this constraint?