Question

1. Let S be a set with n elements. By definition, the power set P(S) is the set whose elements are the subsets of S.

a. Find the order of P(S).

b. Suppose that 0 ≤ r ≤ n. How many elements of P(S) contain exactly r elements?

c. Write a clear justification (that you can understand and explain to yourself!) for the following combinatorial proof:

  P(S) = n union r = 0 {A ⊆ S | |A| = r } ⇒ 2ⁿ = n sigma r = 0 ( n r).

2. All 14 moms of kids on a soccer team in suburban NY own the same Honda midsize SUV, but 4 of the cars are white, 3 are black and 7 are blue. How many different ways can the cars line up in the parking lot?

3. Six dice are rolled. What is the probability of getting exactly 2 ones, 3 twos and 1 six?

4. A multiple choice exam has 10 questions. Each question has 4 choices from which to select the correct answer. Suppose the responses are chosen randomly, with uniform probability. Find the probability of each of the following events:
   a. exactly 3 correct answers; b. at least one correct answer; c. more than 8 correct answers.

5. A classical music lover has 8 chamber music tracks (C), 15 operas (O) and 20 symphonies (S). Suppose that 3 of the chamber music pieces, 4 of the operas and 7 of the symphonies were composed by Mozart. a. If 6 tracks are chosen, find the probability that 2 come from each type: C, O, S. b. If 2 pieces are chosen from each type C, O, S, find the probability that all 6 of the choices were composed by Mozart. c. If 2 pieces are chosen from each type C, O, S, what is the probability that none of the 6 were composed by Mozart?

6. In 1654, the French mathematician Blaise Pascal (a founder of probability theory, famous for Pascal’s triangle and also for building one of the first mechanical computers) was asked about the probability of getting (6,6) at least once in 24 throws two dice. Can you figure it out?

Answer 1

Answer: