Chapter 26 Probability
26.5
Permutations
A permutation is an arrangement of items in a specific order. If a problem asks how many ways
can you arrange 6 books on a bookshelf, it is asking how many permutations there are for 6 items.
Example 9:
Ron has 4 items: a model airplane, a trophy, and autographed football, and a toy
sports car. How many ways can he arrange the 4 items on a shelf?
Solution:
Ron has 4 choices for the first item on the shelf. He then has 3 choices left for
the second item. After choosing the second item, he has 2 choices left for the
third item and only one choice for the last item. The diagram below shows the
permutations for arranging the 4 items on a shelf if he chose to put the trophy
first.
1st item
2nd item
3rd item
4th item
Count the number of permutations if Ron chooses the trophy as the first item. There are 6 permutations. Next, you could construct a pyramid of permutations choosing the model car first.
That pyramid would also have 6 permutations. Then, you could construct a pyramid choosing
the airplane first. Finally, you could construct a pyramid choosing the football first. You would
then have a total of 4 pyramids each having 6 permutations. The total number of permutations is
6 × 4 = 24. There are 24 ways to arrange the 4 items on a bookshelf.
You probably don’t want to draw pyramids for every permutation problem. What if you want to
know the permutations for arranging 30 objects? Fortunately, mathematicians have come up with
a formula for calculating permutations.
For the above problem, Ron has 4 items to arrange. Therefore, multiply 4 × 3 × 2 × 1 = 24.
Another way of expressing this calculation is 4!, stated as 4 factorial. 4! = 4 × 3 × 2 × 1.
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