1D Permutations 1. AMUSEMENT PARKS Seven friends are waiting to ride the new roller coaster. In how many ways can they board the ride, once it is their turn? SOLUTION: There are 7 choices for the first student, 6 choices for the second student, 5 choices for the third student, 4 choices for the fourth student, 3 choices for the fifth student, 2 choices for the sixth student, and then only 1 choice for the last student. So, there are 7 · 6 · 5 · 4 · 3 · 2 · 1 or 5,040 different possible orders. 2. COMMITTEES In how many ways can a president, vice president, and secretary be randomly selected from a class of 25 students? SOLUTION: There are 25 choices for the president, 24 choices for the vice president, and 23 choices for the secretary. So, there are 25 · 24 · 23 = 13,800 choices in all. 3. DVDS You have five seasons of your favorite TV show on DVD. If you randomly select two of them from a shelf, what is the probability that you will select season one first and season two second? SOLUTION: There are 5 · 4 = 20 ways to choose two shows in order. The probability of choosing one of the possible orders is . 4. PASSWORDS A password consists of four letters, none of which are repeated. What is the probability that a person could guess the entire password by randomly selecting the four letters? SOLUTION: There are 26 choices for the first letter, 25 choices for the second letter, 24 choices for the third letter, and just 23 choices for the last letter. So, there are 358,800 possible passwords in all. The possibility that someone guesses the password in one try is . 5. CONTESTS In the Battle of the Bands contest, in how many ways can the four participating bands be ordered? SOLUTION: There are 4 choices for the first band, 3 choices for the second band, 2 choices for the third band, and then only 1 choice for the last band. So, there are 4! = 4 · 3 · 2 · 1 or 24 different possible orders. 6. CODES A garage door code has 5 digits. If no digit is repeated, how many codes are possible? SOLUTION: There are 10 numbers for the first digit, 9 numbers for the second digit, 8 numbers for the third digit, 7 numbers for the fourth digit and 6 numbers for the fifth digit. So, the number of possible outcomes is: 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 = 30,240 7. LETTERS How many permutations are possible of the letters in the word friend ? SOLUTION: There are 6 different letters. There are 6 choices for the first letter, 5 choices for the second letter, 4 choices for the third letter, 3 choices for the fourth letter, 2 choices for the fifth letter, and just 1 choice for the last letter. So, there are 6! = 6 · 5 · 4 · 3 · 2 · 1 or 720 different possible permutations. eSolutions Manual - Powered by Cognero 8. NUMBERS How many different 3-digit numbers can be formed using the digits 9, 3, 4, 7, and 6? Assume no number can be used more than once. Page 1 SOLUTION: 1D There are 6 different letters. There are 6 choices for the first letter, 5 choices for the second letter, 4 choices for the third letter, 3 choices for the fourth letter, 2 choices for the fifth letter, and just 1 choice for the last letter. So, Permutations there are 6! = 6 · 5 · 4 · 3 · 2 · 1 or 720 different possible permutations. 8. NUMBERS How many different 3-digit numbers can be formed using the digits 9, 3, 4, 7, and 6? Assume no number can be used more than once. SOLUTION: There are 5 different digits. There are 5 choices for the first digit, 4 choices for the second digit, and 3 choices for the third digit. So, there are 5 · 4 · 3 or 60 different possible numbers. 9. CAPTAINS The members of the Evergreen Junior High Quiz Bowl team are listed. If a captain and an assistant captain are chosen at random, what is the probability that Walter is selected as captain and Mi-Ling as co-captain? SOLUTION: choices of There are 10 choices for the captain and 9 choices for the co-captain. So, there are captain/co-captain in all. The probability of one of these occurring is . 10. BASEBALL Adriano, Julián and three of their friends will sit in a row of five seats at a baseball game. If each friend is equally likely to sit in any seat, what is the probability that Adriano will sit in the first seat and Julián will sit in the second seat? SOLUTION: st nd There are 5 choices for the 1 seat and 4 choices for the 2 seat. So, there are for the first two seats. The probability of one of these occurring is possible arrangements . 11. GAMES Alex, Aiden, Dexter, and Dion are playing a video game. If they each have an equally likely chance of getting the highest score, what is the probability that Dion will get the highest score and Alex the second highest? SOLUTION: nd There are 4 choices for the highest scorer and 3 choices for the 2 highest. So, there are arrangements for the first two places. The probability of one of these occurring is possible . 12. BLOCKS A child has wooden blocks with the letters shown. Find the probability that the child randomly arranges the letters in the order TIGER. eSolutions Manual - Powered by Cognero SOLUTION: There are Page 2 possible orderings of the letters. The probability of any one such arrangement SOLUTION: nd There are 4 choices for the highest scorer and 3 choices for the 2 highest. So, there are arrangements for the first two places. The probability of one of these occurring is 1D Permutations possible . 12. BLOCKS A child has wooden blocks with the letters shown. Find the probability that the child randomly arranges the letters in the order TIGER. SOLUTION: possible orderings of the letters. The probability of any one such arrangement There are being randomly chosen is eSolutions Manual - Powered by Cognero . Page 3