COSC 208 (Spring 2009)HW 6 Due: 05/04/09

In the questions below suppose that a “word” is any string of seven letters of the alphabet, with repeated letters allowed.

  1. How many words begin with A or B or end with A or B?
  2. How many words begin with a vowel and end with a vowel?
  3. How many words begin with a vowel or end with a vowel?

In the questions below suppose you have a class with 30 students — 10 freshmen, 12 sophomores, and 8 juniors.

  1. In how many ways can you put all 30 in a line?
  2. In how many ways can you put all students in a line so that the freshmen are first, the sophomores are in the middle, and the juniors are at the end?
  3. In how many ways can you get a committee of 7?
  4. In how many ways can you get a committee of 4 freshmen and 3 sophomores?

In the questions below (8 – 12) suppose you have 30 books (15 novels, 10 history books, and 5 math books). Assume that all 30 books are different. In how many ways can you

  1. In how many ways can you put the 30 books in a row on a shelf?
  2. In how many ways can you get a bunch of four books to give to a friend?
  3. In how many ways can you get a bunch of three history books and seven novels to give to a friend?
  4. In how many ways can you put the 30 books in a row on a shelf if the novels are on the left, the math books are in the middle, and the history books are on the right?
  5. A factory makes automobile parts. Each part has a code consisting of a letter and three digits, such as C117, O076, or Z920. Last week the factory made 60,000 parts. Prove that there are at least three parts that have the same serial number.
  6. A computer randomly prints three-digit codes, with no repeated digits in any code (for example, 387, 072, 760). What is the minimum number of codes that must be printed in order to guarantee that at least six of the codes are identical?

In the questions below suppose you have 40 different books (20 math books, 15 history books, and 5 geography books).

  1. You pick one book at random. What is the probability that the book is a history book?
  1. You pick one book at random. What is the probability that the book is not a geography book?
  2. You pick two books at random, one at a time. What is the probability that both books are history books?
  3. You pick two books at random, one at a time. What is the probability that the two books are from different disciplines?

In the questions below an experiment consists of picking at random a bit string of length five. Consider the following events:

E1: the bit string chosen begins with 1;

E2: the bit string chosen ends with 1;

E3: the bit string chosen has exactly three 1s.

  1. Find p(E1E3).
  2. Find p(E3E2).
  3. Find p(E2E3).