EXAM 2
Name
You are required to show your work on every problem in this test
No credit will be given for unsupported answers.
You may use your calculators on this test if you so desire.
Read instructions FULLY before starting a problem.
GOOD LUCK!
Prob. 1 (10 pts) Answer the following questions by writing the
word "true" or the word "false" in the blank precedidng the problem.
a.
If S is any set then the empty set may or may not be a subset of S.
b.
c.
C(12, 10) = 132
d.
If you were asked to find out how many ways there are to arrange three models
of cars in the showroom, if the dealer has 12 models of cars to choose from, you
would use a combination.
e.
Prob.2 (10 pts) Suppose we have the following Tableau's. For each Tableau
circle the pivot element.
Prob. 3 (7 pts)
Let 
, 
and
a. Find 
.
b. Find 
.
c. Find 
.
d. Is B a subset of C? Why or why not?
e. Find 
.
f.
g. Is 
an element in D? Why or why not?
Prob. 4 (5 pts) Set up, but do not solve the following problem.
(Make sure you actually write down the initial tableau)
The Wayland Company manufactures two models of its twin-size futons,
standard and deluxe, in two locations, Houston and San Antonio. The
maximum output at the Houston factory is 600 per week, whereas the maximum output at
the San Antonio factory is 400 per week. The profit per futon for
standard and deluxe models manufactured in Houston is $30 and $20,
respectively. The profit per futon for standard and deluxe models manufactured
in San Antonio is $34 and $18 respectively. For a certain week, the company
has received an order for 600 standard models and 300 deluxe models.
If prior commitments dictate that the number of deluxe models manufactured
in San Antonio may not exceed the number of standard models manufactured
there by more than 50, find how many of each model should be manufactured
at each location so as to satisfy the order and at the same time maximize
Wayland's profit.
Prob. 5 (5 pts) Solve the following linear programming problem.
Minimize C = 2x - y + 3z subject to:
Prob. 6 (9 pts) Some counting problems.
a. A feed store sells 38 brands of bird seed. Twenty-five of these brands
contain millet and thirty-two of the brands contain sunflower seeds.
How many brands contain both Millet and Sunflower seeds?
b. Suppose you wish to visit a relative in Chicago and you know that
you have three distinct routes to your local airport, there are five airlines
flying from this airport into Chicago and there are five routes from there to
your relatives house.
How many different ways are there for you to travel from here to there?
c. Suppose I have a jar which contains 14 red, 20 green and 9 black
marbles. Draw a marble and record its color then replace the marble in the
jar and draw another one and record its color.
How many ways are there to draw a red marble first followed by either
a red marble or a black marble.
Prob. 7 (10 pts) The math club, which has 40 members, wants to form
a committee to plan some social activities.
a. How many ways are there to randomly select a committee of 10 people
from among the math club members?
b. Now that we have formed the committee we need officers for it. How many different ways are there to pick a president, secretary and treasurer from among these 10 people?
Prob. 8 (10 pts) The blue prize box contains 15 super balls and 9 whistles. The red prize box contains 11 super balls and 13 whistles. After a visit to the doctor a child selects a prize box at random then randomly selects a prize from that prize box. What is the probability that a whistle is picked from the red box?
Prob. 9(6 pts) A jar contains 9 red marbles and 14 green marbles.
Draw a marble and record its color then put the marble back into the jar and
draw another marble and record its color.
a. What is the sample space of this experiment?
b. What is the probability that the second marble is green?
Prob. 10 (5 pts) Five people are to pick a number between one and ten. What is the probability that at least two of them chose the same number?
Prob. 11 (10 pts) What is the probability of drawing a full house if you are dealt 7 cards from a well-shuffled, standard fifty-two card deck?
Prob. 12 (15 pts) After a nuclear disaster the cockroaches have mutated.
Several teams of biologists were sent out to gather data on these mutations.
The ones who managed to return reported that there were 40 diffferent
types of man-hunting cockroaches, 55 types of carrion-eaters and 30 types which
were poisonous. Ten of these types were poisonous man-hunters that would also
eat carrion if it was available. Twelve of the non-poisonous types were man-hunters
who would also eat carrion. There were 15 types that were poisonous man-eaters
and 20 types that are strictly non-poisonous, carrion-eaters.
How many types exist that are poisonous, carrion eaters but don't hunt man?
If there were at least 150 types discovered, how many do not fit any of theses categories?
How many non-poisonous, man-hunting types that will not eat carrion exist?