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Section 4

Counting Rules

Zip Codes How many 5-digit zip codes are possible if digits can be repeated? If there cannot be repetitions?

Letter Permutations List all the permutations of the letters in the word MATH.

Speaking Order Seven elementary students are selected to give a 3-minute presentation on what they did during summer vacation. How many different ways can the speakers be arranged?

Visiting Nurses How many different ways can a visiting nurse visit 9 patients if she wants to visit them all in one day?

Quinto Lottery A lottery game called Quinto is played by choosing five numbers each, from 0 through 9. How many numbers are possible? Although repeats are allowed, how many numbers are possible if repeats are not allowed?

Show Programs Three bands and two comics are performing for a student talent show. How many different programs (in terms of order) can be arranged? How many if the comics must perform between bands?

Rolling Dice If five dice are rolled, how many different outcomes are there?

Radio Station Call Letters The call letters of a radio station must have 4 letters. The first letter must be a K or a W. How many different station call letters can be made if repetitions are not allowed? If repetitions are allowed?

Film Showings At the Rogue Film Festival, the director must select one film from each category. There are 8 drama films, 3 sci-fi films, and 5 comedy films. How many different ways can a film be selected?

Secret Code Word How many 4-letter code words can be made using the letters in the word pencil if repetitions are permitted? If repetitions are not permitted?

Passwords Given the characters $A, B, C, H, I, T, U, V,$ $1,2,3,$ and $4,$ how many seven-character passwords can be made? (No repeats are allowed.) How many if you have to use all four numbers as the first four characters in the password?

Automobile Trips There are 2 major roads from city $X$ to city $Y$ and 4 major roads from city $Y$ to city $Z .$ How many different trips can be made from city $X$ to city $Z,$ passing through city $Y ?$

Evaluate each expression.$$\begin{array}{llll}{a .11 !} & {e ._{6} P_{4}} & {i ._{9} P_{2}} \\ {b .9 !} & {f ._ {12} P_{8}} & {j ._{11} P_{3}} \\ {c .}{0 !} & {g ._{7} P_{7}} \\ {d .} {1 !} & {h ., 70}\end{array}$$

Evaluate each expression.$$\begin{array}{llll}{a .} & {6 !} & {e .} & {_{9} P_{6}} \\ {b .} & {11 !} & {f .} & {_{11} P_{4}} \\ {c .} & {2 !} & {g .} & {_{8} P_{0}} \\ {d .} & {9 !} & {h .} & {_{10}P_{2}}\end{array}$$

Sports Car Stripes How many different 4-color code stripes can be made on a sports car if each code consists of the colors green, red, blue, and white? All colors are used only once.

Manufacturing Tests An inspector must select 3 tests to perform in a certain order on a manufactured part. He has a choice of 7 tests. How many ways can he perform 3 different tests?

Endangered Amphibians There are 9 endangered amphibian species in the United States. How many ways can a student select 3 of these species to write a report about them? The order of selection is important.

Inspecting Restaurants How many different ways can a city health department inspector visit 5 restaurants in a city with 10 restaurants?

Word Permutation How many different 4-letter permutations can be written from the word hexagon?

Cell Phone Models A particular cell phone company offers 4 models of phones, each in 6 different colors and each available with any one of 5 calling plans. How many combinations are possible?

Free-Sample Requests An online coupon service has 13 offers for free samples. How many different requests are possible if a customer must request exactly 3 free samples? How many are possible if the customer may request up to 3 free samples?

Ticket Selection How many different ways can 4 tickets be selected from 50 tickets if each ticket wins a different prize?

Movie Selections The Foreign Language Club is showing a four-movie marathon of subtitled movies. How many ways can they choose 4 from the 11 available?

Task Assignments How many ways can an adviser choose 4 students from a class of 12 if they are all assigned the same task? How many ways can the students be chosen if they are each given a different task?

Agency Cases An investigative agency has 7 cases and 5 agents. How many different ways can the cases be assigned if only 1 case is assigned to each agent?

Signal Flags How many different flag signals, each consisting of 7 flags hung vertically, can be made when there are 3 indistinguishable red flags, 2 blue flags, and 2 white flags?

Word Permutations How many permutations can be made using all the letters in the word MASSACHUSETTS?

Code Words How many different 9-letter code words can be made using the symbols $\%, \%, \%, \%, \&, \&,$ $\&,+,+?$

Toothpaste Display How many different ways can 5 identical tubes of tartar control toothpaste, 3 identical tubes of bright white toothpaste, and 4 identical tubes of mint toothpaste be arranged in a grocery store counter display?

Book Arrangements How many different ways can 6 identical hardback books, 3 identical paperback books, and 3 identical boxed books be arranged on a shelf in a bookstore?

Letter Permutations How many different permutations of the letters in the word CINCINNATI are there?

Evaluate each expression.$$\begin{array}{ll}{a . _{5} \mathrm{C}_{2}} & { d. _{6} \mathrm{C}_{2}} \\ {\text b._{3} \mathrm{C}_{3}} & {\text e._{6} \mathrm{C}_{4}} \\ {\text c._{7} \mathrm{C}_{4}}\end{array}$$

Evaluate each expression.$$\begin{array}{ll}{\text { a. }_{3} C_{0}} & {\text { d.} _{12}C_{2}} \\ {\text { b. }_{3} C_{3}} & {\text { e. }_{4} C_{3}} \\ {\text { c. }} {\text { gC }_{7}}\end{array}$$

Medications for Depression A researcher wishes her patients to try a new medicine for depression. How many different ways can she select 5 patients from 50 patients?

Selecting Players How many ways can 4 baseball players and 3 basketball players be selected from 12 baseball players and 9 basketball players?

Coffee Selection A coffee shop serves 12 different kinds of coffee drinks. How many ways can 4 different coffee drinks be selected?

Selecting Christmas Presents If a person can select 3 presents from 10 presents under a Christmas tree, how many different combinations are there?

Buffet Desserts In how many ways can you choose 3 kinds of ice cream and 2 toppings from a dessert buffet with 10 kinds of ice cream and 6 kinds of toppings?

Bridge Foursomes How many different tables of 4 can you make from 16 potential bridge players? How many different tables if 4 of the players insist on playing together?

Music Recital Six students are performing one song each in a jazz vocal recital. Two students have repertoires of five numbers, and the others have four songs each prepared. How many different programs are possible without regard to order? Assume that the repertory selections are all unique.

Freight Train Cars In a train yard there are 4 tank cars, 12 boxcars, and 7 flatcars. How many ways can a train be made up consisting of 2 tank cars, 5 boxcars, and 3 flatcars? (In this case, order is not important.)

Selecting a Committee There are 7 women and 5 men in a department. How many ways can a committee of 4 people be selected? How many ways can this committee be selected if there must be 2 men and 2 women on the committee? How many ways can this committee be selected if there must be at least 2 women on the committee?

Selecting Cereal Boxes Wake Up cereal comes in 2 types, crispy and crunchy. If a researcher has 10 boxes of each, how many ways can she select 3 boxes of each for a quality control test?

Hawaiian Words The Hawaiian alphabet consists of 7 consonants and 5 vowels. How many three-letter â€śwordsâ€ť are possible if there are never two consonants together and if a word must always end in a vowel?

Selecting a Jury How many ways can a jury of 6 women and 6 men be selected from 10 women and 12 men?

Selecting Students How many ways can you pick 4 students from 10 students (6 men, 4 women) if you must have an equal number of each gender or all of the same gender?

Investigative Team The state narcotics bureau must form a 5-member investigative team. If it has 25 agents from which to choose, how many different possible teams can be formed?

Dominoes A domino is a flat rectangular block whose face is divided into two square parts, each part showing from zero to six pips (or dots). Playing a game consists of playing dominoes with a matching number of pips. Explain why there are 28 dominoes in a complete set.

Charity Event Participants There are 16 seniors and 15 juniors in a particular social organization. In how many ways can 4 seniors and 2 juniors be chosen to participate in a charity event?

Automobile Selection An automobile dealer has 12 small automobiles, 8 mid-size automobiles, and 6 large automobiles on his lot. How many ways can two of each type of automobile be selected from his inventory?

DVD Selection How many ways can a person select 8 DVDs from a display of 13 DVDs?

Railroad Accidents A researcher wishes to study railroad accidents. He wishes to select 3 railroads from 10 Class I railroads, 2 railroads from 6 Class II railroads, and 1 railroad from 5 Class III railroads. How many different possibilities are there for his study?

Selecting a Location An advertising manager decides to have an ad campaign in which 8 special calculators will be hidden at various locations in a shopping mall. If he has 17 locations from which to pick, how many different possible combinations can he choose?

Selecting Posters A buyer decides to stock 8 different posters. How many ways can she select these 8 if there are 20 from which to choose?

Test Marketing Products Anderson Research Company decides to test-market a product in 6 areas. How many different ways can 3 areas be selected in a certain order for the first test?

Nuclear Power Plants How many different ways can a government researcher select 5 nuclear power plants from 9 nuclear power plants in Pennsylvania?

Selecting Musicals How many different ways can a theatrical group select 2 musicals and 3 dramas from 11 musicals and 8 dramas to be presented during the year?

Textbook Selection How many different ways can an instructor select 2 textbooks from a possible 17?

DVD Selection How many ways can a person select 8 DVDs from 10 DVDs?

Flight Attendants How many different ways can 3 flight attendants be selected from 11 flight attendants for a routine flight?

Signal Flags How many different signals can be made by using at least 3 different flags if there are 5 different flags from which to select?

Dinner Selections How many ways can a dinner patron select 3 appetizers and 2 vegetables if there are 6 appetizers and 5 vegetables on the menu?

Air Pollution The Environmental Protection Agency must investigate 9 mills for complaints of air pollution. How many different ways can a representative select 5 of these to investigate this week?

Selecting Officers In a board of directors composed of 8 people, how many ways can one chief executive officer, one director, and one treasurer be selected?

Selecting Council Members The presidents, vice presidents, and secretary-treasurers from each of four classes are eligible for an all-school council. How many ways can four officers be chosen from these representatives? How many ways can they be chosen if the president must be selected from the sitting presidents, the vice president from the sitting vice presidents, the secretary from the sitting secretary-treasurers, and the treasurer from everybody whoâ€™s left?

Selecting Coins How many different ways can you select one or more coins if you have 2 nickels, 1 dime, and 1 half-dollar?

People Seated in a Circle In how many ways can 3 people be seated in a circle? 4? n? (Hint: Think of them standing in a line before they sit down and/or draw diagrams.)

Seating in a Movie Theater How many different ways can 5 people- $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D},$ and $\mathrm{E}-$ sit in a row at a movie theater if $(a) \mathrm{A}$ and $\mathrm{B}$ must sit together; $(b) \mathrm{C}$ must sit to the right of, but not necessarily next to, $\mathrm{B} ;(c) \mathrm{D}$ and $\mathrm{E}$ will not sit next to each other?

Poker Hands Using combinations, calculate the number of each type of poker hand in a deck of cards. (A poker hand consists of 5 cards dealt in any order.)$$\begin{array}{l}{\text { a. Royal flush }} \\ {\text { b. Straight flush (not including a royal flush) }} \\ {\text { c. Four of a kind }} \\ {\text { d. Full house }}\end{array}$$

How many different combinations can be made from $(x+2)$ things taken $x$ at a time?

A game of concentration (memory) is played with a standard 52-card deck. How many potential two-card matches are there (e.g., one jack â€śmatchesâ€ť any other jack)?