The calculator function is not reserved solely for very large numbers. In Example 1, 10 P 3 could have been simplified using the calculator function rather than the formula. Whether you use the calculator function instead of the formula often depends on if the question asks for written work to be shown. If not, then using the calculator function only is acceptable.
So far in this lesson, you have seen questions that do not use all the objects in a group. In the notation n P r , this means n ≠r . How does the problem change when n = r ?
In this example, you are using the digits from 1 to 9, which are all distinct objects. In other words, they are not identical. The question, then, is How many ways can you arrange nine distinct objects? By looking at the flow chart, you can see that, when the objects are not identical, n P n = n! . For this example, there are 9 P 9 = 9! = 362 880 ways to arrange the nine digits. |
Can you arrive at this answer in a different way? Try using the formula n P r .
You get the same result! It turns out the n P n is just a special case of n P r , when n = r . Although you can use n P r for this question, it is more efficient to remember that n P n = n !.
How does the solution change when you answer yes to the question Are some of the objects in the group identical?
By looking at the flowchart, you can see that the number of permutations from a set of n objects, in which a are identical, another b are identical, another c are identical, and so on is found using the formula:
The numerator in this formula is the same as n P n because it is an adaptation of n P n . In this formula, a!b!c! ... adjusts n P n to account for the number of identical objects. You will see how a!b!c! ... are used in Example 4.
You may notice that
P
does not have subscripts such as
n
P
n
and
n
P
r
. This is because there is no calculator function to use in lieu of
.