# Third Midterm Exam

### 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.