Explain that a permutation is a set of objects from which you are going to select a certain subset of objects. For example, assume there are 20 students lining up and you have five candy bars of different sizes to offer the first five in line, and the first person will receive the largest, the second the next largest and so on. To figure out how many different arrangements of students could fill the first five slots, you notate the problem as P(20, 5). In other words, how many permutations of five are possible out of a group of 20?
Ask how many students could fill the first slot and elicit the understanding that any of the 20 could get the top slot. Then there are only 19 left to fill the second, 18 for the third, 17 for the fourth and 16 for the fifth and so on.
Explain that the notation P(20,5) means that you start with the whole group, 20 in this case and multiply by each next lower number until you have the same number of factors as the number of selections. Since there are five prize slots available, you will multiply 20 x 19 x 18 x 17 x 16 = 1,860,480 possible arrangements of prize-winning students.
Pass out five playing cards to each student. Ask one student to figure out how many arrangements are possible if she draws just two cards without repeating the order in which the two cards are drawn. For example, the two of hearts and five of diamonds is one possible draw. If she draws two of hearts and then five of diamonds again, it does not count as a separate permutation. However, if she first draws the five of diamonds and then two of hearts, that does count. Ask each student or group to list out all the possible permutations of their card set. Try selecting three at time or starting with a larger set for an extra challenge.
You are commissioned by the president of a newly discovered planet to create a striped tri-color planet flag. You have red, yellow, orange, green, blue and purple cloth. How many possible color arrangements can you create with the available materials? Remember that orange-blue-purple is different than purple-orange-blue. The same three colors can have several possible arrangements. Draw several possibilities and choose your favorite.
You are signing up for classes for the new school year and need to create a schedule to ensure that you get in all your core classes as well as some electives. There are seven periods to fill. Math, language arts, science, history and physical education are required. You may choose two electives from choir, band, art, home economics and a foreign language. How many possible schedule combinations can you create? Is it possible to create a schedule that does not meet the requirements? Discuss why some permutations may be possible but not allowed because of other factors such as academic requirements.