Download Math 356 - Second Exam Choose 5 of the 7 problems.

Math 356 - Second Exam
Prof. Caughman - Thursday, July 31, 2014
NAME
Choose 5 of the 7 problems.
(*You must do problems 1 and 2).
problem points
1*
20
2*
20
3
20
4
20
5
20
6
20
7
20
total
score
100
Show all of your work.
Note: Proofs are graded for clarity and rigor, so show all of your
steps and justify all your inferences. Some tips:
(1). Work neatly. Write in complete sentences.
(2). State any assumptions you are making.
(3). Clearly state the conclusions you draw from your argument.
1*. Induction
Prove by induction that for any natural number n,
−12 + 22 − 32 + · · · + (−1)n n2 = (−1)n
n(n + 1)
.
2
2*. Pigeonhole Principle
Forty ping-pong balls, numbered 1 to 40, are placed in a vat. How
many must be drawn to ensure:
(i). A pair whose sum is 40?
(ii). A pair whose product is 12?
(iii). At least 5 odd numbers?
3. Equivalence Relations
(a). Let X = {1, 2, 3, 4, 5} and recall that X × X denotes the set of
ordered pairs of elements of X. Define a relation ∼ on X × X by
(a, b) ∼ (c, d) if and only if a − b = c − d.
(a). Show that ∼ is an equivalence relation on X × X.
(b). List one member of each equivalence class.
4. Counting
(a). How many different rearrangements can be made of the letters in
TALLAHASSEE?
(b). How many can be made if the two L’s must appear next to each
other, and the two E’s cannot appear next to each other?
5. Binomial Coefficients
(a). How many subsets with an odd number of elements does a set with
8 elements have? How many subsets have an even number of elements?
(b). Find the coefficient of x7 in (2x + 3)10 .
(c). Describe all triples of integers a > b > c > 0 for which
a b
a
=2
.
b
c
c
6. Relations
For a, b ∈ N, define a relation R by aRb whenever a ≤ 2b. Determine whether the relation R is reflexive, symmetric, transitive, and/or
antisymmetric. Prove your answers.
7. Counting and Probability.
(a). How many solutions are there to the equation
n1 + n2 + n3 + n4 = 23
if n1 , n2 , n3 , n4 are integers satisfying n1 ≥ 2, n2 ≥ 3, n3 ≥ 4, and
n4 ≥ 5?
(b). What is the probability that a randomly chosen subset of the set
{1, 2, 3, 4, 5, 6, 7, 8, 9}
contains exactly two odd numbers?