# Counting

## How many bit strings are there of length eight?

Always remember to have the definitions, axioms, lemmas, theorems, and corollaries necessary to solve the exercises you want to solve at hand (or memorized). Without that, you won’t be able to solve the exercises. Relevant definitions for this exercise: THE PRODUCT RULE: “Suppose that a procedure can be broken down into a sequence of two […]

## How many different three-letter initials with none of the letters repeated can people have?

Relevant definitions for this exercise: THE PRODUCT RULE: “Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways to do the second task, then there are

## There are four major auto routes from Boston to Detroit and six from Detroit to Los Angeles. How many major auto routes are there from Boston to Los Angeles via Detroit?

Relevant definitions for this exercise: THE PRODUCT RULE: “Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways to do the second task, then there are

## A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in three sizes for each sex. How many different types of this shirt are made?

Relevant definitions for this exercise: THE PRODUCT RULE: “Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways to do the second task, then there are

## How many different three-letter initials are there that begin with an A?

Relevant definitions for this exercise: THE PRODUCT RULE: “Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways to do the second task, then there are

## Six different airlines fly from New York to Denver and seven fly from Denver to San Francisco. How many different pairs of airlines can you choose on which to book a trip from New York to San Francisco via Denver, when you pick an airline for the flight to Denver and an airline for the continuation flight to San Francisco?

Relevant definitions for this exercise: THE PRODUCT RULE: “Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways to do the second task, then there are

## A multiple-choice test contains 10 questions. There are four possible answers for each question

Let’s first write relevant definitions for this exercise: THE PRODUCT RULE: “Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways to do the second task,

## How many bit strings of length n, where n is a positive integer, start and end with 1s?

As always, we should start with knowing the rules and/or definitions. THE PRODUCT RULE: “Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways to do

## An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?

Let’s review the sum rule. THE SUM RULE: “If a task can be done either in one of n1 ways or in one of n2 ways, where none of the set of n1 ways is the same as any of the set of n2 ways, then there are n1 +n2 ways to do the task.”

## How many bit strings of length ten both begin and end with a 1?

Below are the rules that we can use for this type of exercise. THE PRODUCT RULE: Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways