Math
211
Third
Hour Exam
Name
__________________________
Friday, November
15
100 pts.
1.
(15
pts.) Suppose that
and that 
. Calculate
A
+ B
A*B
(matrix product)
2. (10
pts.) Consider the graph
Label
the vertices in any order you want (1,2,3,4,5) and write the incidence matrix
of this graph.
3. (10
pts.) We assign marbles at random to
five boxes. How many marbles do we need
to ensure that at least one box has at least four marbles?
4. (5
pts.) Briefly state the technique for
proving a statement using mathematical induction
5 (15
pts.) Use mathematical induction to
show that the sum of the first n odd numbers is n2 . That is, use
induction to show that
Using
the following steps:
a. State and prove the base case
(more)
b. State the inductive step as it applies to
this problem.
c. What is the inductive hypothesis?
d. Complete the proof by proving the inductive
step.
6. Some
counting questions (5 pts. each)
a. In how many ways can we construct a sequence
of five digits (0 – 9)? What principle
of counting are we using?
b. In how many ways can we select 4 marbles from
a bag of 10 marbles.
c. How many solutions do we have to the equation
where each of the
three variables is an integer greater than or equal to 0 (but with no other
restrictions on the variables)?
d. To get to Aberystwyth, one travels by
airplane to
e. What is the coefficient of 
in the expansion of 
?
f. Use Pascal’s identity to express C(10, 7) in
terms of C(9,something)
(more)
g. How many subsets are there of a set with 6
elements?
h. How many integers are there between 1 and 10
which are either even or a prime? What
principle of counting are we using?
7. (5 pts.) Say something appropriate about one of the following:
a)
Fibonacci
(Leonardo of
b)
Gabriel
Lamé
c)
John
McCarthy
d)
G.
Lejeune Dirichlet
e)
Blaise
Pascal (say something more than to say that he was responsible for Pascal’s
identity (problem (f) above))