MATH2800 Introduction to Discrete Structures

Second Exam, Tuesday, March 20, 2007.

You may use one sheet of handwritten notes, but no other sources. The exam consists of six questions, and lasts fifty minutes. Please work all six problems. Please show all work clearly and in reasonable detail. Answers without appropriate supporting work or requested explanations may not receive full credit. No calculators are allowed. Please ring your section below:

1. (25 points)

   A crate with twelve slots contains five apples, three pears, and four oranges. All fruits are indistinguishable within type.

   1. (9 points) In how many ways can the twelve pieces of fruit be placed in the crate? (Express your answer in factorials and/or powers of integers.)

   2. Jing picks three pieces of fruit from the crate, one after the other, without replacement.
      1. (8 points) What is the probability that Jing ends up with three apples, assuming each piece of fruit in the crate is equally likely to be picked? (Express your answer as a fraction whose numerator and denominator are relatively prime.)
      2. (8 points) What is the probability that Jing picks at least one orange, assuming each piece of fruit in the crate is equally likely to be picked? (Express your answer as a fraction whose numerator and denominator are relatively prime.)

2. (10 points)

   What is the coefficient of $x^{13}$ in $(x-3)^{20}$? (Express your answer in factorials and/or powers of integers.)

3. (15 points)

   The Dow Jones Industrial Average (DJIA) and the Nasdaq Composite (NAS) are two stock indices, and the returns of the two indices are related. Assume the NAS has an annual return of at least 12% with probability $1/2$. Assume that if the NAS has an annual return of at least 12% then the DJIA has an annual return of at least 10% with probability $9/10$, and if the NAS has an annual return of less than 12% then the DJIA has an annual return less than 10% with probability $7/10$. In a given year, we observe that DJIA has a return of 13%. What is the probability that the NAS had an annual return of at least 12% in the given year? (Express your answer as a fraction whose numerator and denominator are relatively prime.)

4. (20 points)

   If $E$ and $F$ are independent events, prove or disprove that $\bar{E}$ and $F$ are necessarily independent events.

5. (20 points) A committee of eight students is to be chosen from 10 seniors, 8 juniors, 11 sophomores, and 9 freshmen.
   1. (10 points) For this part, assume the members of each class are indistinguishable. In how many ways could the committee be formed? (Express your answer as an integer.)
   2. (10 points) It is required that the committee contain at least one upperclassman (senior or junior) and at least one lowerclassman (sophomore or freshman). In how many ways could the committee be formed, assuming the members are distinguishable? (Express your answer in factorials and/or powers of integers.)

6. (10 points) The number of ways to pick $k$ items from $n$ when the order is important is $\frac{n!}{(n-k)!}$. The number of ways to pick $k$ items from $n$ when the order is unimportant is $\frac{n!}{k!(n-k)!}$. Explain why these two numbers differ by a factor of $k!$