Finite - Sample Spaces & Fundamental Counting Principle Problems

(FCP AND and OR) Liam's parents won't allow MORE THAN two pets. In a pet shop he decides between 20 hamsters and 10 turtles. How many choices does he have, if he wants to buy? 1. 2 turtles 2. 2 hamsters 3. 1 hamster AND 1 turtle 4. 1 hamster OR turtle 5. 1 hamster 6. 1 turtle 7. none (they all look sick to him)

1. 10x9 = 90 2. 20x19 = 380 3. 20x10 = 200 4. 20 + 10 = 30 5. 20 6. 10 7. 1

(Sample Space) 1. Eight rooms must be painted either white or beige 2. Six books are arranged from left to right 3. A math quiz has 5 multiple choice questions each with four options (a-d) 4. A combination lock had four digits which can each range from 0-9 5. A club must choose meeting times. Possible days are Tuesday, Wednesday, or Thursday and possible times are 3pm, 4pm, and 5pm. They can choose from 10 classrooms.

1. 2x2x2x2x2x2x2x2 or 2^8 = 256 2. 6x5x4x3x2x1 or 6! = 720 3. 4x4x4x4x4 or 4^5 = 1024 4. 10x10x10x10 or 10^4 = 10,000 5. 3x3x10 or 3^2x10 = 90

(FCP Practice 1) 1. In how many ways can a president and a secretary be chosen from a group of 6? 2. In how many ways can a president, vice president, and secretary, be chosen from a group of 10 people? 3. How many 3 letter code words can be made from the letters of the word PROBLEM if: -3a. letters CANNOT be repeated ; 3b. letters CAN be repeated? 4. Five roads connect Cheer City and Glumville. Starting at Cheer City, how many different ways can Happy drive to Glumville and return by the SAME road? How many different round trips can he make if he wishes to return by a DIFFERENT road then he took to Glumville? 5. Filipe has 3 ties, 10 shirts and 4 pairs of trousers. How many different outfits can he wear? Assume that he wears one of each kind of article. 6. Minnesota license plate numbers consist of 3 letters followed by 3 digits (for example AFF033). How many different plates could be issued? 7. Papa's pizza place offers 3 choices of salad, 20 kinds of pizza, and 4 different deserts. -7a. How many different 3-course meals can one order? ; 7b. How many different 2-course meals can one order? 8. How many 3, 4, or 5 letter words can be formed from the letters of the word CREAM (no repetition)? 9.How many 4 letter words can be formed using the letters of the word FACTOR (no repetion) if: -9a. starting with R? ; 9b. with only two consonants ; 9c. with vowels and consonants alternating? ; 9d. with vowels in two middle positions? 10. 3 brothers and 3 sisters are lined up to be photographed. How many arrangements are there: -10a. altogether ; 10b. with brothers and sisters alternating positions ; 10c. with the 3 sisters standing together 11. Sue wants to arrange 5 history books, 4 math books, and 3 psychology books on a shelf. In how many ways can she do this if... -11a. there are no restrictions as to arrangement ; 11b. she puts the 5 history books on the left, the 4 history books in the middle and the 3 psychology books on the right ; 11c. she insists only that books on the same subject be together?

1. 6x5 = 30 2. 10x9x8 = 720 3a. 7x6x5 = 210 3b. 7x7x7 or 7^3 = 343 4. Same way: 5x1 ; Different: 5x4 = 20 5. 3x10x4 = 120 6. 26^3x10^3 = 17,576,000 7a. 3x20x4 = 240 ; 7b. 3x20 + 3x4 + 20x4 = 152 8. 3 letter: 5x4x3 + 4 letter: 5x4x3x2 + 5 letter: 5x4x3x2x1 = 300 9a. 1x5x4x3 ; 9b. 4x3x2x1 9c. 4x2x3x1 or 2x4x1x3 = 2(24) = 28 ; 9d. 4x2x1x3 10a. 6x5x4x3x2x1 or 6! = 720 ; 10b. 2(36) = 72 10c. 4(6x6) = 144 11a. 12! = 479,001,600 ; 11b. 5!x4!x3! = 17,280 ; 11c. 3(5!x4!x3!) = 103,680

(FCP Practice 2) 1. How many different 5-digit numbers exist? 2. How many 5-digit numbers exist if repetition is NOT allowed? 3. How many 5-digit numbers END and BEGIN with an ODD number? 4. How many 5-digit numbers end and begin with an ODD number, if repetition is NOT allowed? 5. How many 5-digit numbers contain only ODD numbers from the 1st to the 5th digit? 6. How many ODD 2-digit positive integers less than 50 are there? 7. How many EVEN 2-digit positive integers greater than 60 are there? 8. How many EVEN 3-digit positive integers can be written using the digits 1,3,4,5,6? 9. How many positive integers less than 1,000 can be written using the digits 3,4,5,6? 10. A username consists of 4 letters followed by 4 digits. How many usernames can be created? 11. A username consists of 4 letters followed by 4 digits, but no letter or digit can be repeated. How many such usernames can be created? 12. A username consists of 4 letters followed by 4 digits, but no adjacent symbols can be repeated. How many such usernames can be created?

1. 9x10x10x10x10 or 9x10^4 2. 9x9x8x7x6 = 27,216 3. 5x10x10x10x5 or 5^2x10^3 = 25,000 4. 5x8x7x6x4 5. 5^5 = 3,125 6. 4x5 = 20 7. 4x5 = 20-1 = 19 8. 5x5x2 = 5^2x2 = 50 9. 4x4x4 + 4x4 + 4 = 84 10. 26x26x26x26x10x10x10x10 or 26^4x10^4 11. 26x25x24x23x10x9x8x7 = 1,808,352,000 12. 26x25x25x25x10x9x9x9 = 26x25^3x10x9^3

Fundamental Counting Principle (FCP): Two such principles, the first one involving multiplication (AND) and the second addition (OR).

1. If one selection can be made in m ways and for each of these a second selection can be made in n ways, then the number of ways the two selections can be made is mxn. (Multiplication = AND) 2. If the possibilities being counted can be grouped into mutually exclusive cases, then the total number of possibilities is the sum of the number of possibilities in each case. (Addition = OR

(FCP Practice 3) 1. CA license plate numbers consist of 1 number then 3 letters followed by 3 numbers -1a. How many different CA plates could be issued if repetition is allowed? ; 1b. if NO repetition is allowed ; 1c. If no vowels and no zeros are allowed ; 1d. if every license plate has to start with a 1 and a C 2. How many 5 number license plates can be made using the digits 1,2,3,4,5,6,7,8 if an ODD digit must come first and... -2a. repetition is allowed ; 2b. repetition is NOT allowed 3. How many 6 number license plates can be made using the digits 1,2,3,4,5,6,7,8 if ODD and EVEN digits must alternate AND -3a. repetition is allowed ; 3b. repetitions are NOT allowed 4. Some license plates have 2 letters (26 choices) followed by 4 digits (10 choices). How many license plates can be created if... -4a. letters and digits can be repeated ; 4b. letters can be repeated but not digits ; 4c. if neither letters nor digits can be repeated; 4d. three letters followed by 4 digits is allowed, all with repetition

1a. 10x26x26x26x10x10x10 or 26^3x10^4 ; 1b. 10x26x25x24x9x8x7 ; 1c. 9x21x21x21x9x9x9 or 21^3x9^4 ; 1c. 1x1x26x26x10x10x10 or 26^2x10^3 2a. 4x8x8x8x8 or 4x8^4 ; 2b. 4x7x6x5x4 = 3,360 3. 4x4x4x4x4x4 and 4x4x4x4x4x4 or 4^6x2 ; 4x4x3x3x2x2 = 2(4!x4!) = 1,152 4a. 26x26x10x10x10x10 or 26^2x10^4 ; 4b. 26x26x10x9x8x7 or 26^2x10x9x8x7 ; 4c. 26x25x10x9x8x7 ; 4d. 26x26x26x10x10x10x10 or 26^3x10^4

(FCP Review) 1. No special conditions 2. Without repetition 3. Even 4. Even, without repetition 5. Begin with an even, and end with an odd number 6. Begin with an even, and end with an odd number, never repeating any number 7. Using only 1,2,3,5,6 8. What is the largest number of this set 9. Even, using only 1,2,3,5,6 10. What is the smallest number of this set?

2 digit ; 3 digit ; 4 digit 1. 9x10 ; 9x10x10 ; 9x10x10x10 2. 9x9 ; 9x9x8 ; 9x9x8x7 3. 9x5 ; 9x10x5 ; 9x10x10x5 4. 8x5 ; 8x8x5 ; 8x8x7x5 5. 4x5 ; 4x10x5 ; 4x10x10x5 6. 4x5 ; 4x8x5 ; 4x8x7x5 7. 5x5 ; 5x5x5 ; 5x5x5x5 8. 66 ; 666 ; 6,666 9. 5x2 ; 5x5x2 ; 5x5x5x2 10. 12 ; 112; 1,112

(FCP And and OR) You have $6 for lunch and can choose ONLY one item. The choice is between 3 kinds of burgers and 4 kinds of pizzas. How many choices are there?

3 (burgers) + 4 (pizzas) = 7 choices

(FCP AND and OR) You have $20 for lunch. The choice is between 3 kinds of burgers and 4 kinds of pizzas. You can choose ONE OF EACH. How many choices do you have?

3 x 4 = 12

(Sample Space) A basket contains an apple, an orange and a peach. You randomly pick two fruit at a time.

AO, AP; OA, OP; PA, PO 3x2 = 6

(Sample Space) A coin is tossed 3 times

H - H, T - H, T T - H, T - H, T 2^3 = 8

(Sample Space) A spinner can land on either red, blue or green. You spin twice

R - R,B,G B - R,B,G G - R,B,G 3x3 = 9

(Sample Space) A coffee shop offers small, medium and large sizes of French, Italian and American Roast

SF, SI, SA; MF, MI, MA; LF, LI, LA 3x3 = 9

(FCP continued) In how many different ways can an 8-question true-false test be answered: a. if every question must be answered b. if it is all right to leave questions unanswered?

a. 2x2x2x2x2x2x2x2 or 2^8 b. 3x3x3x3x3x3x3x3 or 3^8

(FCP continued) In how many ways can you select 4 cards, one after another, from a 52 card deck: a. if the cards are returned to the deck after being selected b. if the cards are not returned to the deck after being selected

a. 52x52x52x52 or 52^4 = 7, 311,616 b. 52x51x50x49 = 6,497,400

(FCP continued) A bookshelf contains 6 different algebra books, 5 different geometry books, and 3 different trigonometry books. In how many ways can you select: a. any one of the math books b. one of each of the three types of math books

a. 6 + 5 + 3 = 14 math books b. 6x5x3 = 90 if one from each group