CMPSCI 250: Introduction to Computation
Third Midterm Exam
David Mix Barrington
18 November 2004
Directions:
- Answer the problems on the exam pages.
- There are five problems
for 100 total points.
Probable scale is A=90, C=60.
- If you need extra space use the back of a page.
- No books, notes, calculators, or collaboration.
- You may use exponents, falling exponents, "choose"
notation, and/or factorials to express numerical answers.
However, for full credit naturals less than 100
must be computed out fully and expressed in decimal.
Q1: 20 points
Q2: 20 points
Q3: 20 points
Q4: 25 points
Q5: 15 points
Total: 100 points
- Question 1 (20):
Sgt. Preston of the Yukon is assembling a team of eight dogs to pull
his sled. The candidate dogs are of three breeds: Alaskan, Mixed, and
Siberian. The team will be arranged in a single line beginning with the
lead dog.
- (a,5) Assuming that he has at least eight of each breed available, how
many choices does he have for which breed occupies which position in the team?
(For example, he might pick "Alaskan, Mixed, Siberian, Mixed, Mixed,
Alaskan, Siberian, Siberian".)
- (b,5) Again assuming he has at least eight of each breed available, how
many choices does he have for how many dogs of each breed are on his
team? (In the example above, he has picked "two Alaskan, three Mixed, and
three Siberian".)
- (c,5) Now suppose he has chosen two Alaskans, two Mixed, and four
Siberians for the team. How many choices does he have for which breed goes
in which position? (Hint: First count the ways to place the two Alaskans, then
count the ways to place the two Mixed among the remaining positions.)
- (d,5) Again assuming that he has picked his eight dogs, in how many ways
can he select which dogs go in the first four positions? (For example, he
might pick "dog 7, dog 3, dog 8, dog 1" in that order.)
- Question 2 (20):
In Game C the player throws a single fair six-sided die k times
in sequence. She wins if she completes the k throws without ever throwing
the same number twice in a row. Prove by induction for all positive
naturals k that the probability of winning Game C is exactly
(5/6)k-1.
- Question 3 (20):
Let n and k be naturals and let Σ be an alphabet of size n. The
language S(n,k) is the set of all strings over Σ that (a) contain
exactly k distinct letters, (b) have each letter at most twice, and (c) have
each two occurrences of the same letter, if any, next to each other. For
example, with Σ = {a,b}, S(2,1) = {a, aa, b, bb} and S(2,2) =
{ab, abb, aab, aabb, ba, baa, bba, bbaa}.
- (a,10) Determine the number of elements in S(5,3).
- (b,10) Determine the number of elements in S(n,k) for any naturals
n and k.
- Question 4 (25):
This problem deals with two more dice games. In Game D the player
throws k six-sided dice at once. She wins $6 for every die that comes up
six. In Game E she again throws k six-sided dice, but this time she wins
$6 if at least one six comes up, and nothing otherwise.
- (a,10) Calculate the expected value of Game D, as a function of k.
- (b,15) Calculate the expected value of Game E, as a function of k.
(Hint: It may be easiest to compute the probability that the player
loses Game E.)
- Question 5 (15):
Define a directed multigraph G to have vertex set {1,2,3} and the
following adjacency matrix:
1 0 2
1 1 0
0 0 1
- (a,5) Draw the directed multigraph G.
- (b,10) Compute the number of one-step, two-step, four-step, and
eight-step paths from vertex 2 to vertex 3 in G.
Last modified 16 December 2004