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2018 Midterm


To choose 5 peoms from 3 books containing 30 poems each, which is 90 peoms in total. How many ways can this be done if:
1. there are no restrictions?
2. at least 1 poem must be selected from each book?


Provide a 2-column proof:
If A\subset B\land C\subset D then A\land C \subset B\land D
(Do not use a membership table or Veen diagram.)


Use a 2-column, rules of inference proof to find who killed Boromir:
a. If Elrond was in the wizard's council, then Sarumon killed Boromir.
b. Either Galarel or Elrond was in the wizard's council.
c. If Frodo was in the forest, then Ugluk killed Boromir.
d. If Frodo was not in the forest, then Elrond was not in the wizard's council.
e. If Ugluk killed Boromir, hekilled Boromir with a sword.
f. If Galadriel was in the wizard's council, then Lurtz killed Boromir.
g. Boromir was killed by an arrow.


How many was can 60 different programmers be assigned to exactly 1 project each, if there are 5 different projects and:
1. there are no other restrictions?
2. each project must have 12 programmers?


Given the theorem: If a quadrilateral (a 4 sided figure) is a rectangle, then it has 2 pairs of parallel sides. Can I conclude:
1. If a quadrilateral has 2 pairs of parallel slides, then it is a rectangle.
2. If a quadrilateral does not have 2 pairs of parallel sides, then it is not a rectangle.
Please explain your answer using concepts learned in this class.


Prove using induction:
for n is an odd, positive integer, 8|(n^2-1).


When she is about to leave the coffee shop, Ms. Right sees she has a penny, a nickle, a dimes, a quarter, and a half-dollar. In how many ways can she leave a tip if:
1. there are no restrictions
2. she leaves at least 1 coin?


A professor has 22 books on computer science that cover topics of (A) compilers (B) data structures and (C) operating systems. If there are 8 books that cover compilers, 13 books that cover data structures, and 13 books that cover operating systems; 5 books cover compilers


Write 22 as base 2, base 8 and base 10.