Homework 7

CS 280 - Spring 2002

Due: Friday, March 29

Part A

  1. A die is tossed 300 times.  Let Yi be the random variable that counts the number of times that i is rolled where i is one of 1, 2, 3, 4, 5, or 6.  
    1. Find E(Y6).
    2. Find V(Y6).
    3. Find E(Y1+Y2+Y3).
    4. Find V(Y1+Y2+Y3).  (Hint: Think Bernoulli trials; depending on how you solve this, you may need to be careful about independence.)
       
  2. Continuing from the problem 1, let X be the random variable defined as X = Y1+2Y2+3Y3+4Y4+5Y5+6Y6.
    1. Find E(X).
    2. Find V(X).  (Hint: Think Bernoulli trials, but be careful about independence.)

Part B

  1. Four people are dividing among themselves 4 identical apples, 3 identical bananas, and 2 identical pears.  How many ways can this be done?  Note that it's possible for a person to receive no fruit at all.  Also note that the people are distinguishable; in other words, a solution in which the first person gets both pears is considered to be different from any solution in which the second person gets both pears.
     
  2. You have 5 Computer Science books, 6 Math books, and 8 Romance novels.  
    1. Assume that the books are distinguishable.  How many ways can these books be arranged on a shelf so that no two Computer Science books are adjacent?
    2. Assume that the books within a category are indistinguishable (i.e., all the CS books are alike, all the Math books are alike, and all the Romance novels are alike, although it's easy to tell the CS books from the Math books from the Romance novels).  How many ways can these books be arranged on a shelf so that no two Computer Science books are adjacent?

Part C

  1. How many positive integers less than 4445 are divisible by 2, 3, 5, or 7?
     
  2. Two necklaces are considered to be the same if the necklaces can be rotated and/or turned over so that one looks like the other.  Assume that there are no knots or other "landmarks".
    1. How many different necklaces can be made using a red bead, a green bead, a blue bead, and 20 identical yellow beads?
    2. How many different necklaces can be made using three identical black beads and 20 identical yellow beads?