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The Fundamental Counting Principle provides us with a tool that allows us to calculate the number of outcomes possible in many situations. What if the situation is a bit more complex? For many situations, the order that we complete a task does not matter. Ordering milk, bacon, and scrambled eggs in that order is the same as ordering bacon, scrambled eggs, and milk. In this case the order that we make our choices wouldn't matter, but there are many situations in which the order that we do things does make a difference.
3 things are important to remember:
1. A permutation is an arrangement or sequence of selections of objects from a single set.
2. Repetitions are not allowed. Equivalently the same element may not appear more than once in an arrangement.
3. The order in which the elements are selected or arranged is significant.
A permutation is a specific order or arrangement of a set of objects or items. What if you wish to call someone on the phone? If I make the call, the order that I punch in the numbers matters so this is an example of a permutation. A good question to ask when deciding if your arrangement is a permutation is"DOES ORDER MATTER?" If yes, then you are dealing with a permutation. For example, if you ordered an ice cream sundae and they put the cherry in first, then the chocolate sauce, and then the ice cream, you would probably would not be happy with that particular ice cream sundae. You would likely prefer that they put the ice cream in first, then the chocolate sauce, and then put the cherry on top. Clearly each sundae had the same three ingredients, but they were quite different from one another. Each order that we can make the ice cream sundae is called a permutation.
There is a simple formula for figuring out how many permutations exist when 'r' objects are selected from a set of 'n' objects. The left side of the equation can be read "n P r", just as it looks or "n Permutations of size r".
[Figure2]
Recall that the exclamation point is a factorial. For example,
5! = 5 x 4 x 3 x 2 x 1 = 120
Also, be sure to find the permutations command on your calculator.
In our ice cream sundae discussion, 'n' would be 3 because there are 3 items to select from and 'r' would also be 3 because we are going to select all three items. Using the permutations formula, this would be
In other words, there are 6 different orders that the ice cream sundae could be made. Note that 0! is equal to 1.
Example 1
Suppose you are going to order an ice cream cone with two different flavored scoops. You are going to take a picture of your ice cream cone for use in the school newspaper. The ice cream shop has 5 flavors to choose from; chocolate, vanilla, orange, strawberry, and mint. How many different ice cream cone photos are possible?
Solution
The first question to ask is "Does Order Matter?". If it does, then we are dealing with a permutation question. In this case, the order does make a difference. A chocolate on top of vanilla cone looks different than a vanilla on top of chocolate cone. We have five flavors to pick from, so n=5. We are going to select 2 flavors so r=2.
There 20 different permutations of ice cream cones we could order. The notation representing this situation,
can be read as "Five 'P' Two" or "Five permutations of size Two". Be sure to perform this calculation using your calculator as well.
In the example above, you could have also found your answer using the Fundamental Counting Principle. There were 5 choices for the 1st flavor and then only 4 choices for the 2nd flavor. There are 5 x 4 = 20 ice cream cones possible.
[Figure3]
Example 2
Give the value of
by using the formula for permutations. Verify your solution on your calculator.
Solution
Example 3
Decide whether each of the situations below involves permutations.
a) A five-card poker hand is dealt from a deck of cards.
b) A cashier must give 3 pennies, 2 dimes, a 5 dollar bill, and a 10 dollar bill back as change for a purchase.
c) A student is going to open a padlock that has a three number combination.
d) A child has red, blue, green, yellow, and orange color crayons and will be coloring a rainbow using each color one time.
Solution
a) The order you get your five cards for a poker hand does not matter. If one of your cards was the ace of spades, it didn't matter if it was the first card or the last card dealt.
b) The order that the cashier gives you $15.23 in change does not matter as long as the total is $15.23.
c) The order you put in the three numbers for the combination makes a difference. If the correct combination is 12-27-19, the padlock will not open if you enter 19-12-27 even though the same three numbers are used.
d) The order that the child colors the rainbow does make a difference. The color pattern red, blue, green, orange, yellow will look different than green, blue, red, yellow, orange.
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