1. business and industrial

8.1 The Multiplication Principle; Permutations
Ex 1) A certain combination lock can be set to open to any 3­letter sequence.
a. How many sequences are possible?
b. How many sequences are possible if no letter is repeated?
Ex 1b) A combination lock can be set to open to any 4­digit sequence. How many sequences are possible?
How many sequences are possible if no digit is repeated?
Ex 2) Morse code uses a sequence of dots and dashes to represent letters and words. How many sequences are possible with at most 3 symbols?
Ex 3) An ancient Chinese philosophical work known as the I Ching (Book of Changes) is often used as an oracle from which people can seek and obtain advice. The philosophy describes the duality of the universe in terms of two primary forces: yin (passive, dark, receptive) and yang (active, light, creative). Yin and yang can be represented by the following symbols.
Yin
Yang
These lines can be written on top of one another in groups of three, aka trigrams. The trigram is called Tui, the Joyous, and has the image of a lake.
a. How many trigrams are there altogether?
b. The trigrams are grouped together, one on top of the other, in pairs known as hexagrams. Each hexagram represents one aspect of the I Ching philosophy. How many hexagrams are there?
Ex 4) A teacher has 5 different books that he wishes to arrange side by side. How many different arrangements are possible?
Ex 5) Suppose the teacher in Example 4 wishes to place only 3 of the 5 books on his desk. How many arrangements of 3 books are possible?
Ex 6) Find the following.
a. The number of permutations of the letters A, B, and C.
b. The number of permutations if just 2 of the letters A, B, and C are to be used.
Ex 7) In a recent election, eight candidates sought the Democratic nomination for president. In how many ways could voters rank their first, second, and third choices?
Ex 8) A televised talk show will include 4 women and 3 men as panelists.
a. In how many ways can the panelists be seated in a row of 7 chairs?
b. In how many ways can the panelists be seated if the men and women are to be alternated?
c. In how many ways can the panelists be seated if the men must sit together, and the women must also sit together?
d. In how many ways can one woman and one man from the panel be selected?
Ex 9a) How many ways are there to arrange the letters of the word "STEP"? What about "STEEP"?
Distinguishable Permutations
Ex 9b) In how many ways can the letters in the word "MISSISSIPPI" be arranged?
Ex 10) A student buys 3 cherry yogurts, 2 raspberry yogurts, and 2 blueberry yogurts. She puts them in her dormitory refrigerator to eat one a day for the next week. Assuming yogurts of the same flavor are indistinguishable, in how many ways can she select yogurts to eat for the next week?
Ex 10b) A student has 4 pairs of identical blue socks, 5 pairs of identical brown socks, 3 pairs of identical black socks, and 2 pairs of identical white socks. In how many ways can he select socks to wear for the next two weeks?
8.2 Combinations
Ex 1a) How many committees of 3 people can be formed from a group of 8 people?
Ex 1b) How many committees of 5 people can be formed from a group of 8 people?
Ex 2) Three lawyers are to be selected from a group of 30 to work on a special project. a. In how many different ways can the lawyets be selected?
b. In how many ways can the group of 3 be selected if a certain lawyer must work on the project?
c. In how many ways can a nonempty group of at most 3 lawyers be selected from these 30 lawyers?
Ex 3) A salesman has 10 accounts in a certain city.
a. In how many ways can he select 3 accounts to call on?
b. In how many ways can he select at least 8 of the 10 accounts to use in preparing a report?
Ex 4) For each problem, tell whether permutations or combinations should be used to solve the problem.
a. How many 4­digit code numbers are possible if no digits are repeated?
b. a sample of 3 light bulbs is randomly selected from a batch of 15. How many different samples are possible?
c. In a baseball conference with 8 teams, how many games must be played so that each team plays every other team exactly once?
d. In how many ways can 4 patients be assigned to 6 different hospital rooms so that each patient has a private room?
Ex 5) A manager must select 4 employees for promotion; 12 employees are eligible.
a. In how many ways can the 4 be chosen?
b. In how many ways can 4 employees be chosen (from 12) to be placed in 4 different jobs?
Ex 6) Five cards are dealt from a standard 52­card deck.
a. How many such hands have only face cards?
b. How many such hands have a full house of aces and eights (3 aces and 2 eights)?
c. How many such hands have exactly 2 hearts?
d. How many such hands have cards of a single suit?
Ex 7) Suppose 2 cans of soup are to be selected from 4 cans on a shelf: noodle (N), bean (B), mushroom (M), and tomato (T). Illustrate the difference between permutations and combinations by making two tree diagrams of these soups.
8.3 Probability Applications of Counting Principles
Ex 1) The Environment Protection Agency is considering inspecting 6 plants for environmental compliance: 3 in Chicago, 2 in Los Angeles, and 1 in Hew York. Due to a lack of inspectors, they decide to inspect 2 plants selected at random, 1 this month and 1 next month, with each plant equally likely to be selected, but no plant is selected twice. What is the probability that 1 Chicago plant and 1 Los Angeles plant are selected?
Ex 2) From a group of 22 nurses, 4 are to be selected to present a list of grievances to management.
a. In how many ways can this be done?
b. One of the nurses is Lori Hales. Find the probability that Hales will be among the 4 selected.
c. Find the probability that Hales will not be selected.
Ex 3) When shipping diesel engines abroad, it is common to pack 12 engines in one container. Suppose that a company has received complaints from its customers that many of the engines arrive in nonworking condition. To help solve this problem, the company decides to make a spot check of containers after loading. The company will test 3 engines from a container at random; if any of the 3 are nonworking, the container will not be shipped until each engine in it is checked. Suppose a given container has 2 nonworking engines. Find the probability that the container will not be shipped. Ex 4) In a common form of the card game poker, a hand of 5 cards is dealt to each player from a deck of 52 cards.
a. How many different hands are possible?
b. What is the probability of dealing a hand containing only hearts, called a heart flush?
c. What is the probability of dealing a flush of any suit (5 cards of the same suit)?
d. What is the probability of dealing a full house of aces and eights (3 aces and 2 eights)?
e. What is the probability of dealing any full house (3 cards of one value, 2 of another)?
Ex 5) Each of the letters w, y, o, m, i, n, and g is placed on a separate slip of paper. a slip is pulled out and its letter is recorded in the order in which the slip was drawn. This is done four times.
a. If the slip is not replaced after the letter is recorded, find the probability that the word "wing" is formed.
b. If the slip is replaced after the letter is recorded, find the probability that the word "wing" is formed.
Ex 6) Suppose a group of n people is in a room. Find the probability that at least 2 of the people have the same birthday.
Ex 7) Ray and Nate are arranging a row of fruit at random on a table. They have 5 apples, 6 oragnes, and 7 lemons. What is the probability that all fruit of the same kind are together?