Permutations and Combinations
Instructional Notes
Math A30
Permutations and Combinations
Instructional Notes
It may be necessary to review some of the underlying principles of set theory before beginning the study of this section. The concepts of set, subset, and mutually exclusive should be understood, at least intuitively.
The fundamental counting principles are those which deal with the combining of the numbers of outcomes in given situations.
FCP 1 (The Multiplication Principle)
If the first part of a procedure can be performed in m ways, and the second part of the procedure can be performed in n ways, then the procedure can be carried out in mn ways.
FCP 2 (The Addition Principle)
If one task can be performed in m ways, and a second, mutually exclusive task can be performed in n ways, then the number of ways of performing either task is given by m+n ways.
Students should work in groups to determine the answers to several questions involving these counting principles. Discussion should be encouraged within the groups, so that group members are satisfied that their group answers are suitable. Group answers should be summarized. (PSVS)
Students should be given the definition of a permutation, and shown some examples of permutations (e.g; 3-letter nonsense syllables composed of the letters a, b, and c).
The idea of a permutation of a complete set of elements and its relation to the multiplication principle should be explored by having groups work on selected examples or exercises.
When students are comfortable with their understanding, the factorial notation could be introduced, and formalized. Calculators, or computers should be employed to do calculations of larger permutations, and perhaps to run simulations.
Note the special case of 0! = 1. Have students try this on their calculators as well.
Students can be given some relatively simple examples to work on in their groups. The group answers should be shared, with ensuing discussion leading to a more uniform method of approaching these situations. (An example that they might work on is: list all the different two-letter syllables that could be formed from the word 'star'.)
The formula should be developed, and further examples done to illustrate the use of the formula. Also, utilize calculators and computers.
Students could work in groups to determine how many different ways they could arrange three objects in a line, if all were different. They could then progress to the situation where two of the objects were identical. Extend this to have them determine the number of ways to arrange four objects in a line, if two were identical (then three identical).
At this point, they could be asked to conjecture a result for the general case. Have them predict the result with five objects, two of which are identical. Extend this to more difficult situations, such as five objects, of which three are identical to each other, and the remaining two are identical to each other.
Manipulatives, such as bingo chips, algebra tiles, dice, could be useful for this activity.
For circular permutations, students might be given a situation to explore individually, in pairs, or small groups. Use of manipulatives such as small figurines, coins, stamps, and the like, may be useful for some, but these could also be done by assigning alpha-numeric symbols to positions on a circle. The problem assigned might be to determine in how many ways three (then four, five, and n) objects can be arranged in a circle. Students should be instructed that there is no `fixed' starting position, and to note each permutation. After determining the results for a few of these, students should be expected to determine the formula for this type of permutation. { (n-1)! }
Some exercises that utilize this formula can be assigned for practice.
The special case of the `key-ring' type of permutation can also be dealt with in this section. Because some objects, such as coloured beads on a necklace, or keys, can be arranged in a circle (key-ring) without regard to whether they are right side up, the number of permutations is given by the formula
This special case can be demonstrated by/to students using beads/keys on a necklace/ring.
Students should be presented with the definition of combination, and the major differences of combinations and permutations should be pointed out; that for combinations, the order of arrangement is not important.
Groups could work on a set of exercises designed to build an understanding of the underlying principles of combinations.
These exercises should be chosen to reflect or model real-world situations, and manipulatives should be available to the groups to use in their explorations.(NUM)
Students should be reminded of the fundamental counting principles before beginning this topic.
Groups could work on the solutions of example problems that reflect this situation. Group answers, and the rationale for these answers, should be shared with the class.(COM)
The examples chosen should try to reflect real-world situations insofar as possible.
An introductory problem might be posed to the students to work on cooperatively. An example might be similar to the following:
An art student is instructed to make a collage of 2 different triangles, 3 different quadrilaterals, and two pentagons. In the supply material provided, there are six different triangles, 5 different quadrilaterals, and 4 different pentagons. In how many different ways may the student choose the materials required?