Math A30
Permutations and Combinations
Adaptations Open this page in its own window

A.1

Student groups could be instructed to share some real-life instances where fundamental counting principles are used (e.g.; generation of postal codes, telephone numbers, types of hamburgers available at local booths or concessions, licence plates, etc.).

Some calculations might be done with these examples to determine if the needs of the community are being served by these systems. (E.g.: Will the present system of phone numbers be suitable if the community expands in the future? How could this be determined?)

The teacher may wish to introduce questions dealing with the intersection and union of sets at this point. These may be introduced using counting arguments or overlapping sets. A number of examples should be done to illustrate concepts such as the principles of inclusion and exclusion.

inclusion/exclusion formula using A and B

Example: 93 students attended a school dance. 62 students purchased hotdogs while 45 purchased a soft drink, and 8 had neither. How many students had both a hotdog and a soft drink?

A.2

Students could be asked to generate permutations for their own groups to solve. Each example would have to be reflected upon to see if it fits the criteria of a permutation of a set of objects, before the calculations are carried out.

Students should use the multiplication principle to find the number of ways to form a queue of length n from n people. They would fill in the first position, then the second, and so on.

A.3

Students could be asked to determine real-world situations in which permutations of this type are encountered. Then the number of permutations of each example can be determined by using formula, calculator, or computer. The teacher should have some examples ready in case students are not able to generate their own examples.(CCT)

Students could be asked to find the number of ways to form a queue of length r from n people using the multiplication principle.

Students could be instructed to answer the following: If you could do one operation from a list of 20! every second, how many years would it take to perform all the operations ?

A.4

These exercises are meant to be completed using the formula in the resource texts. It is a variation of n factorial divided by n subscript 1 times n subscript 2 etc. where n... represent the number of times an element appears in the arrangement.

This topic can be extended by introducing many other real-world applications. Students could be asked to generate these in groups, and set up related questions to solve. (IL)

A.5

The questions given to the students can be adapted quite readily for better students by introducing restrictions similar to number 5 in the examples/activities column.

For example, students could attempt questions such as:
In how many ways may we seat three couples (men and women) at a table, if men and women must be seated alternately?

In how many ways can we seat five people in a circle, if we do not want Bill and Bernie to sit next to each other?

In how many different ways may we arrange seven objects in a circle, if two of the objects are identical?

A.6

This topic could also be studied by using a queue of length r taken from n people. Pick r from n people and then line them up in r! different ways. Then we have subscript n P subscript r = subscript n C subscript r times r factorial and therefore, subscript n C subscript r = n factorial divided by (n minus r)factorial times r factorial

Note that the two above methods are equivalent for repetitions; for example, the number of permutations of the letters of MISSISSIPPI is
eleven factorial divided by 4 factorial times 4 factorial times 2 factorial times 1 factorial

which can be seen as 11C4×7C4×3C2 by picking places for each set of letters and using the multiplication principle. Try others to determine if this equivalence is general for repetitions.

A.7

The groups can extend their skills and knowledge of this topic by researching examples of combinations present in other real-world situations; environmental, hereditary, games, politics (such as the constitutional debate over Senate reform), economics, and others. These could be done as simply listing situations where combinations occur, or mini-reports could be expected reflecting writing skills and some computations as well.

Students could work on a specific situation such as determining the number of ways it is possible to choose a dozen doughnuts from five varieties, if at least one of each variety must be chosen? Two of each?

Student discussion on this type of problem may lead to other concepts (such as the binomial theorem) which could be briefly discussed at this time, if student interest warrants.