1.10 Problem Solving with Permutations and Combinations 1) 12 football players stand in a circular huddle. How many different arrangements of the players are there? or = 3 991 680 2) How many 3-digit numbers can Doug form using only the numbers 1 to 7 if the number 2 must be included? # with 2 = total # - # without 2 = 7 x 7 x 7-6 x 6 x 6 = 7 3-6 3 = 127 3) The Greek alphabet contains 24 letters. How many different Greek-letter fraternity names can Jess create using either 2 or 3 letters? (Repetitions are allowed) = # with 2 letters + # with 3 letters = 24 2 + 24 3 = 14400 4) In how many ways can 11 players be seated on the team bench so that Lindsay and Rachael are not seated next to each other? # ways L&R not together =total # ways - # ways L&R together =11! - 10!2! =32659200 5) In how many ways can 4 men and 4 women be seated around a circular table if each man must be flanked by two women? (Hint: arrange the men first and then the women) 3!4! or = 144
6) Find the number of ways you can choose at least 1 piece of fruit from a basket containing 4 apples, 5 bananas, 2 cantaloupes and 3 pears. use = 5 x 6 x 3 x 4-1 = 359 since at least one 7) In how many ways can you select a chairman, treasurer and secretary from a board of directors with 8 members? P(8,3) = 336 8) If 1000 people enter a contest in which there is a first prize, second prize and third prize, in how many ways can the prizes be given? P(1000,3) = 997 002 000 9) Chris and Shannon are starting out on their evening run. Their route always takes them 8 blocks east and 5 blocks north to Mike s apartment building. Chris likes to vary the path each night. How many different possible routes does Chris have? = 1287 or apply Pascal s Method 10) Jen and Lindsay are in charge of assigning rooms to the players on the team. In how many ways can they assign the 12 basketball players to 4 triple rooms? = 369 600
11) Danielle joins Cameron on his trip to the giant auction sale late in the afternoon. There are only 5 items left to be sold. How many different purchases could Cameron make? 2 5-1 = 31 or = 31 12) If Isabelle, Glen, Leah and Patricia play doubles matches in tennis, how many matches are necessary if every player has every other player as a partner? 3! different teams but 2 teams / game, so = 3 or = 3 13) A group of 25 students is flying to Akron, Ohio for their grad trip. There are 25 seats available on the plane, 6 of which are first class. Alex and Heather won a draw and must sit in first class. Rachel, Jenna and David are socially conscious and refuse to sit in first class. With these restrictions in mind, in how many ways can the students be divided between first class and economy? 6 for First Class, 19 econ - 2 FC seats taken so only 4 left and 3 econ seats taken so only 16 left: = 4845 14) Richard wants to skateboard over to visit his friend Cathy who lives six blocks away. Cathy s house is 2 blocks west and 4 blocks north of Richard s house. Each time Richard goes over, he likes to take a different route. How many different routes are there for Richard if he only travels west and north? = 15 or apply Pascal s Method 15) How many different sums of money can Justin form from one $2 bill, three $5 bills, two $10 bills and one $20 bill?
16) A 12-volume encyclopedia is to be placed on a shelf. In how many ways can Amanda arrange them such that they are in an incorrect order? # incorrect = total # - # correct = 12! - 1 = 479 001 599 17) There are 12 questions on an examination, and Margie must answer 8 questions including at least 4 of the first 5 questions. How many different combinations of questions could she choose to answer? = 210 18) The 6 members of the yearbook staff sit around the circular table in their office. How many different seating arrangements are there of this group of people? or 5! = 120 19) There are 8 runways at the regional airport. Pilots Jen, Alex and Jesse are each bringing their planes in for a landing at approximately the same time. In how many ways can air-traffic control sergeant Matt assign the planes to different runways? P(8,3) = 336 20) A committee of 3 teachers is to select the winner from among 15 students nominated for a special award. The teachers each make a list of their top 3 choices in order. The lists have only 1 name in common, and the name has a different rank on each list. In how many ways could the teachers have made the lists? # choices for common name x # arrangements of common name x # arrangements remaining = 15 x 3! x P(15,6) = 324 324 000 P(15,1) x P(14,2) X P(12,2) x P(10,2) x 3! = 324 324 000