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PART 1 MODULE 4
THE FUNDAMENTAL COUNTING PRINCIPLE
EXAMPLE 1.4.
Plato is going to choose a three-course meal at his favorite restaurant. He must choose one item from each of the following three categories. First course: Tofu Soup (TS); Seaweed Salad (SS) Second course: Steamed Tofu (ST); Baked Tofu (BT); Fried Tofu (FT); Third course: Tofu Cake (TC); Tofu Pie (TP); Seaweed Delight (SD) How many different three-course meals are possible? Solve this problem by listing every possible 3-course meal. SOLUTION We will list every possible 3-course meal:
- TS-ST-TC 2. TS-ST-TP 3. TS-ST-SD 4. TS-BT-TC
- TS-BT-TP 6. TS-BT-SD 7. TS-FT-TC 8. TS-FT-TP
- TS-FT-SD 10. SS-ST-TC 11. SS-ST-TP 12. SS-ST-SD
- SS-BT-TC 14. SS-BT-TP 15. SS-BT-SD 16. SS-FT-TC
- SS-FT-TP 18. SS-FT-SD We see that there are 18 different three-course meals. However, there is an easier way to get at this answer. Notice that in order to choose a three-course meal, we need to make three decisions:
- Choose item for first course: There are 2 options.
- Choose item for second course: There are 3 options
- Choose option for third course: There are 3 options. Now observe that (2)(3)(3) = 18. This is not a coincidence. It is an illustration of the Fundamental Counting Principle.
The Fundamental Counting Principle (FCP) To determine the number of different outcomes possible in some complex process:
_1. Analytically break down the process into separate stages or decisions.
- Count the number of_ options _that are available at each stage or decision.
- Multiply together all of the numbers from Step 2 above._ EXAMPLE 1.4. How many different three course meals are possible if Plato has already decided that he won't have Baked Tofu and he will have Seaweed Delight? SOLUTION We will use the Fundamental Counting Principle.
- Choose item for first course: 2 options
- Choose item for second course: 2 options (since he won't choose Baked Tofu)
- Choose item for third course: 1 option (since it has already been decided that he will choose Seaweed Delight). (2)(2)(1) = 4 different 3-course meals The primary advantage to the use of the Fundamental Counting Principle becomes apparent when we have more complicated decision processes. For example: EXAMPLE 1.4. Referring to the situation in the previous example, suppose that the restaurant becomes quite successful and so the operators decide to expand their menu. Now there are 5 items for the first course, 7 items for the second course, 4 items for the third course, and a new fourth course is added (the seltzer course, in which we choose between Bromo and Alka ). How many four-course meals are possible? SOLUTION To choose a 4-course meal requires that we make four decisions. There are 5 options, 7 options, 4 options and 2 options, respectively, for each of these decisions. (5)(7)(4)(2) = 280 different 4-course meals. It would be very cumbersome to try to solve this problem by listing every possible outcome, since there are so many different outcomes.
EXAMPLE 1.4.
There are 5 guys (including Gomer) on Gomer's bowling team. After the beer frame they will each choose one of the following: Scud , Scud Lite , or Scud Ice. How many outcomes are possible? A. 60 B. 125 C. 15 D. 243 E. 120 EXAMPLE 1.4. In Florida, standard automobile license plate "numbers" used to follow the following scheme: 3 letters -- 2 digits -- 1 letter Examples: JKP47R TRR39S VWN22Y ZQW05Z How many different license plate codes were possible under this scheme?
EXAMPLE 1.4.
Gomer has to take a 5 question true/false quiz, but he hasn't studied. He will guess at each problem. In how many different ways is it possible to answer the quiz questions? How likely is it that he will get a score of 100%? EXAMPLE 1.4. Gomer has to take a 25 question multiple-choice test, but he hasn't studied. He will guess at each problem. For each problem the possible responses are A, B, C, or D. In how many different ways is it possible to answer the test questions? How likely is it that he will get a score of 100%?
EXAMPLE 1.4.
- How many different 4-digit numbers can be formed using the following digits? (Note: the first digit cannot be 0, or else the number would be a 3-digit number). {0, 2, 3, 5, 8}
- How many different 4-digit numbers that are multiples of 5 can we form? EXAMPLE 1.4. Gomer is considering the purchase of a new super-cheap sport/utility vehicle, the Skuzuzi Kamikaze. He must choose a vehicle, taking into account the following options: i. Transmission: 4 - speed standard transmission, 5-speed standard transmission, or automatic transmission; ii. Bumper: steel bumpers, vinyl bumpers or 2x4 boards bolted to the front and back; iii. Top: hard-top, vinyl top convertible, or chicken wire stapled over the roll bar; iv. Funerary accessory: complementary funeral wreath or cremation urn. How many different vehicle option packages are possible?
EXAMPLE 1.4.
Homer, Gomer, Plato, Euclid, Socrates, Aristotle, Homerina and Gomerina form the board of directors of the Lawyer and Poodle Admirers Club. They will choose from amongst themselves a Chairperson, Secretary, and Treasurer. No person will hold more than one position. How many different outcomes are possible? A. 336 B. 24 C. 512 D. 21 EXAMPLE 1.4.13 SOLUTION Choosing a Chairperson, Secretary and Treasurer from among these 8 people requires us to make three decisions. However (unlike in all of the previous examples) these three decisions are not independent. For instance, the choice we make when we select the chairperson affects which options are available when we go to choose the Secretary, since the person selected to be Chairperson cannot also be selected to be Secretary. i. Choose Chairperson: 8 options; ii. Choose Secretary: 7 options (one person has already been chosen to be Chairperson); iii. Choose Treasurer: 6 options (two people have already been chosen to be Chairperson and Secretary, respectively). According to the Fundamental Counting Principle the number of outcomes is: (8)(7)(6) = 336.
EXAMPLE 1.4.
Adam, Beth, Charles, Dawn, and Ernie are the five finalists in the fabulous Clearers Publishinghouse sweepstakes. Through a random selection process, one of them will win a pen-and-pencil set, one of them will win a one-year supply of cat food, and one of them will win a brand new Chevy blazer (that is, a vinyl sports jacket with the words "Chevy" embossed on the back). Nobody will win more than one prize. How many different outcomes are possible? A. 125 B. 15 C. 12 D. 60 EXAMPLE 1.4. The Shakesperean Insult-O-Matic A B dreadful fiend unmanner'd dog murderous hog wrangling cacodemon elvish-mark'd dissembler abortive toad rooting homicide foul hedgehog defused infection of a man cursed villain devilish lump of foul deformity detested devil poisonous minister of hell bunchback'd beggar bottled spider slander of thy mother's heavy womb Suppose we want to form a three-part Shakespearean insult, such as "Thou dreadful , wrangling , minister of hell " by choosing two adjectives from column A and one noun phrase from column B. How many different insults are possible? (Assume, for instance, that " Unmanner'd , elvish-marked , lump of foul deformity " is different from " Elvish-marked , unmanner'd , lump of foul deformity .") The vocabulary comes from Act I of King Richard III.
EXAMPLE 1.4.
Homer is going to buy a 1986 Yugo. He has shopped around, and has found that ' Yugos are available with any combination of the following options: plush Corinthian vinyl interior, padded steering wheel, tinted windows, bucket seats, motor, windshield. How many different option packages are possible? EXAMPLE 1.4. The Egotists' Club has 6 members: A, B, C, D, E, and F. They are going to line up, from left to right, for a group photo. After lining up in alphabetical order (ABCDEF), Mr. F complains that he is always last whenever they do things alphabetically, so they agree to line up in reverse order (FEDCBA) and take another picture. Then Ms. D complains that she's always stuck next to Mr. C, and that she never gets to be first in line. Finally, in order to avoid bruised egos, they all agree to take pictures for every possible left-to-right line-up of the six people. How many different photos must be taken? SOLUTION To arrange the six people in a line we need to make six dependent decisions:
- Choose first person (6 options);
- Choose second person (5 options)
- Choose third person (4 options)
- Choose fourth person (3 options)
- Choose fifth person (2 options)
- Choose last person (1 option) According to the Fundamental Counting Principle the number of outcomes is (6)(5)(4)(3)(2)(1) = 720. Note that the number of ways to arrange 6 people is equal to 6 multiplied all of the positive integers smaller than 6. Likewise it is easy to see that if there were 8 people rather than 6 people, the number of ways to arrange them in a line would be (8)(7) (6)(5)(4)(3)(2)(1) and if there were 10 people instead of 6 or 8 the number of ways to arrange them in a line would be (10)(9)(8)(7) (6)(5)(4)(3)(2)(1) These products are called factorials. (6)(5)(4)(3)(2)(1) is called “6 factorial” and is denoted 6! (8)(7) (6)(5)(4)(3)(2)(1) is called “8 factorial” and is denoted 8! (10)(9)(8)(7) (6)(5)(4)(3)(2)(1) is called “10 factorial” and is denoted 10!
6. Harpo, Groucho, Chico, Zeppo, Gummo and Karl are running in a race. How many different orders of finish are possible? A. 36 B. 64 C. 720 D. 46656 7. Socrates, Euclid, Plato, Aristotle, Diogenes, and Democritus are going to choose from amongst themselves one person to wax the floor and another person to scrub the toilet. How many different outcomes are possible if Diogenes will not scrub the toilet? A. 36 B. 720 C. 30 D. 25 8. Gomer is going to order a pizza for his cat, Fido. The following toppings are available for cat pizzas: sparrow sprinkles, lizard lozenges, mousie munchies, butterfly bits, and beetle bites. He may choose any combination of those toppings (this includes the possibility that he may choose none of them). How many different toppping combinations are possible? A. 32 B. 25 C. 120 D. 3125 9. When Diogenes the used-car salesman gets dressed for work, he has to choose a jacket, shirt, tie, belt, trousers and shoes. He has a plaid jacket, a checkered jacket, and a pink jacket; a white long-sleeved shirt, a white short-sleeved shirt, a green short-sleeved shirt and a yellow long-sleeved shirt; a striped tie, a polka-dotted tie, a checkered tie, and a solid orange tie; a white patent-leather belt and a black patent-leather belt; black trousers, brown trousers, gray trousers, plaid trousers and pin-striped trousers; brown shoes, black patent-leather shoes, white patent-leather shoes, and sneakers. How many different ensembles are possible today if he has already determined that he won’t wear any stripes and he will wear everything that is white patent-leather? A. 288 B. 144 C. 15 D. 16 10. There are eight finalists in the Miss Sopchoppy contest. How many different outcomes are possible if one person will be selected First Runner-Up and another will be Miss Sopchoppy? A. 56 B. 64 C. 16 D. 256 11. Jethro has just purchased a hamster to be a companion for his gerbil, D.W. He is going to name the hamster by choosing a first name and second name from this list of names: Billy, Bobby, Bubba, Brett. The first name may or may not differ from the second name. How many different two-part names are possible? A. 8 B. 12 C. 16 D. 64 12. There are four Gators in a holding cell at the jail. They will be asked to arrange themselves from left to right in a police line-up. How many different line-ups are possible? A. 16 B. 8 C. 24 D. 256
13. Aristotle is going to register for classes next term. He has to choose one class from each of the following categories: I. MGF1107, MAC1105, STA II. REL2000, REL2121, REL III. AFH1000, AMH1000, ASH3100, EUH IV. LIT2020, LIT2081, LIT3 043 How many different combinations of classes are possible if he has already decided that he will take LIT3043 and he won’t take EUH2100 and he won’t take REL2000? A. 108 B. 19 C. 24 D. 18 14. Harpo, Groucho, Chico, Zeppo, Gummo and Karl are the finalists in the fabulous Clearers’ Publishinghouse sweepstakes. Two prizes will be awarded: the Grand Prize and the Even Grander Prize. The prizewinners will be randomly selected (and it is possible that one person may win both prizes). How many outcomes are possible? A. 720 B. 128 C. 30 D. 36 15. Euclid needs to create a password for his internet account. To simplify things, he is going to form a four-letter password by choosing letters from this set: {A, D, E, M, R, S, T}. How many different passwords are possible if the second and fourth letters will be vowels and the other two letters won’t be vowels (repeated letters are possible)? A. 40 B. 100 C. 256 D. 14 16. Socrates is going to buy a new pencil sharpener. He finds that the following six optional accessories are available: titanium bearings, fuel injection, CD rom, chromium alloy blades, air bags, HEPA filter shavings collector. How many different accessory combinations are possible? A. 64 B. 36 C. 720 D. 12
