Learning Resources
Lesson
You can easily complete this investigation on your own with the use of a calculator. The idea is to find an expression to represent permutations using factorials.
Step A is simply a statement of the Fundamental counting principle to write the solution to the problem. However, Step B requires you to write this as a ratio or quotient of factorials; check out the use of factorials as ratios. Complete the remaining steps.
Let's see how this can be applied to counting permutations by reconsidering a version of the problem considered in the previous lesson:
Example
Suppose there are 10 people in your Mathematics 3205 class who write a unit test. Also, suppose each receives a different score on the test. In how many different ways could the 1st, 2nd, and 3rd highest scores have been distributed among the 10 students?
Solution
Since order matters in this problem, what we are counting is the number of permutations of 10 objects taken 3 at a time. There are 10 students, thus 10 ways of assigning the 1st mark, 9 ways of assigning the 2nd mark, and 8 ways of assigning the 3rd mark. By the Fundamental Counting Principle this means there were 10 x 9 x 8 different ways of assigning the mark.
We now wish to express the product 10 x 9 x 8 as the ratio of factorials. This gives:
The numerator of the fraction is obviously the number in the sample set, viz. the 10 students in the class. But what about the denominator? What is its relation to the numbers in the original problem?
Recall there were 10 in the sample set (the number of students) and we were assigning 3 marks in order. Note that 10 - 3 = 7. We could thus write the above expression as:
Thus we can say that the number of permutations of 10 items taken 3 at a time is given by the formula:
It is cumbersome to write "the number of permutations of 10 items taken 3 at a time". Once again mathematics has provided a shorthand way of writing this English expression. The notation used for the above expression is 10P3 and we read this as "10 permute 3".
Some examples of applying this formula and the notation are shown below:
"the number of permutations of 20 items taken 5 at a time"
"the number of permutations of 15 items taken 10 at a time"
Formula for Calculating Permutations
In general, we can say that "the number of permutations of n items taken r at a time" or "n permute r" is given by the formula:
You should now be ready to do the work assigned for this lesson.
Activity
Investigation Questions p.331. Complete 15
Think About p.332
Check Your Understanding p.332 . Complete 16, 17, 18, 19 and 20
When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.
Test Yourself
- In horse racing, the exactor is betting on which horse will finish 1st, which one will finish 2nd, and which one will finish 3rd. In a race with 9 horses, how many different ways are there for picking the exactor?
- How many different 4 letter permutations can be made from the word DEMOGRAPHY?
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