Two fair dice are thrown. What is the probability that the

How many six-digit numbers are there? How many of them contain the digit 5? Note that the first digit of an n-digit number is nonzero.

How many different five-letter codes can be made using a, b, c, d, and e? How many of them start with ab?

The population of a town is 20,000. If each resident has three initials, is it true that at least two people have the same initials?

In how many different ways can 15 offices be painted with four different colors?

In flipping a fair coin 23 times, what is the probability of all heads or all tails?

In how many ways can we draw five cards from an ordinary deck of 52 cards (a) with replacement; (b) without replacement?

Two fair dice are thrown. What is the probability that the outcome is a 6 and an odd number?

Mr. Smith has 12 shirts, eight pairs of slacks, eight ties, and four jackets. Suppose that four shirts, three pairs of slacks, two ties, and two jackets are blue. (a) What is the probability that an all-blue outfit is the result of a random selection? (b) What is the probability that he wears at least one blue item tomorrow?

A multiple-choice test has 15 questions, each having four possible answers, of which only one is correct. If the questions are answered at random, what is the probability of getting all of them right?

Suppose that in a state, license plates have three letters followed by three numbers, in a way that no letter or number is repeated in a single plate. Determine the number of possible license plates for this state.

A library has 800,000 books, and the librarian wants to encode each by using a code word consisting of three letters followed by two numbers. Are there enough code words to encode all of these books with different code words?

How many n m arrays (matrices) with entries 0 or 1 are there?

How many divisors does 55,125 have? Hint: 55,125 = 325372.

A delicatessen has advertised that it offers over 500 varieties of sandwiches. If at this deli it is possible to have any combination of salami, turkey, bologna, corned beef, ham, and cheese on French bread with the possible additions of lettuce, tomato, and mayonnaise, is the delis advertisement true? Assume that a sandwich necessarily has bread and at least one type of meat or cheese.

How many four-digit numbers can be formed by using only the digits 2, 4, 6, 8, and 9? How many of these have some digit repeated?

In a mental health clinic there are 12 patients. A therapist invites all these patients to join her for group therapy. How many possible groups could she get?

Suppose that four cards are drawn successively from an ordinary deck of 52 cards, with replacement and at random. What is the probability of drawing at least one king?

A campus telephone extension has four digits. How many different extensions with no repeated digits exist? Of these, (a) how many do not start with a 0; (b) how many do not have 01 as the first two digits?

There are N types of drugs sold to reduce acid indigestion. A random sample of n drugs is taken with replacement. What is the probability that brand A is included?

Jenny, a probability student, having seen Example 2.6 and its solution, becomes convinced that it is a nearly even bet that someone among the next 22 people she meets randomly will have the same birthday as she does. What is the fallacy in Jennys thinking? What is the minimum number of people that Jenny must meet before the chances are better than even that someone shares her birthday?

A salesperson covers islands A, B, ... , I . These islands are connected by the bridges shown in the Figure 2.3. While on an island, the salesperson takes one of the possible bridges at random and goes to another one. She does her business on this new island and then takes a bridge at random to go to the next one. She continues this until she reaches an island for the second time on the same day. She stays there overnight and then continues her trips the next day. If she starts hertrip from island I tomorrow, in what percent of all possible trips will she end upstaying overnight again at island I ?

In a large town, Kennedy Avenue is a long north-south avenue with many intersections. A drunken man is wandering along the avenue and does not really know which way he is going. He is currently at an intersection O somewhere in the middle of the avenue. Suppose that, at the end of each block, he either goes north with probability 1/2, or he goes south with probability 1/2. Draw a tree diagram to find the probability that, after walking four blocks, (a) he is back at intersection O; (b) he is only one block away from intersection O.

An integer is selected at random from the set {1, 2, . . . , 1, 000, 000}. What is the probability that it contains the digit 5?

How many divisors does a natural number N have? Hint: A natural numberN can be written aspn1 1 pn2 2 pnk k , wherep1, p2, . . . , pk are distinct primes.

In tossing four fair dice, what is the probability of tossing, at most, one 3?

A delicatessen advertises that it offers over 3000 varieties of sandwiches. If at this deli it is possible to have any combination of salami, turkey, bologna, corned beef, and ham with or without Swiss and/or American cheese on French, white, or whole wheat bread, and possible additions of lettuce, tomato, and mayonnaise, is the delis advertisement true? Assume that a sandwich necessarily has bread and at least one type of meat or cheese.

One of the five elevators in a building leaves the basement with eight passengers and stops at all of the remaining 11 floors. If it is equally likely that a passenger gets off at any of these 11 floors, what is the probability that no two of these eight passengers will get off at the same floor?

The elevator of a four-floor building leaves the first floor with six passengers and stops at all of the remaining three floors. If it is equally likely that a passenger gets off at any of these three floors, what is the probability that, at each stop of the elevator, at least one passenger departs?

A number is selected randomly from the set {0000, 0001, 0002, . . . , 9999}. What is the probability that the sum of the first two digits of the number selected is equal to the sum of its last two digits?

What is the probability that a random r-digit number (r 3) contains at least one 0, at least one 1, and at least one 2?