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PERMUTATIONS AND COMBINATIONS
In this unit, you will first examine some probability topics such as the “Fundamental Counting Principal”, permutations, and combinations. Permutations and combinations are different ways to make arrangements out of objects. A permutation is an arrangement of objects in which the order of the arrangement is important to the outcome. A combination is an arrangement of objects where order is not important.
Fundamental Counting Principal
Permutations
Combinations
Fundamental Counting Principle
If there are m ways that one event can occur and n ways that another event can occur, then there are m n ways that both events can occur.
Example 1 : A movie theatre sells popcorn in small, medium, and large containers. Each size is also available in regular or buttered popcorn. How many options for buying popcorn does the movie theatre provide?
There are 6 possible options for buying popcorn at the movie theatre.
small
regular
buttered
Outcomes
small, regular
small, buttered
medium
regular
buttered
large
regular
buttered
Size Flavor
medium, regular
medium, buttered
large, regular
large, buttered
Permutations
Another way to arrange objects is called permutations. A permutation is an arrangement of objects in a specific order. Such arrangements could include the batting order of a softball team, seat assignments in a classroom, or items displayed on a store shelf.
The following is the formula for finding the number of permutations of n objects taken r at a time.
Example : Find the number of ways to listen to 5 CDs from a selection of 12 CDs.
There are 95,040 different ways to listen to 5 CDs from a selection of 12 CDs.
12 5
P
× × × × ×
12 5
P =
12 5
P =
12 P 5 = 95, 040
Permutation of " objects taken at a time" ! ( )! n r
n r n P n r
Combinations
A well-planned meal or balanced diet gives you all the nutrients you need each day. To plan a balanced diet, you need to select foods from each of the main food groups. The food pyramid below is a practical tool to help you make food choices that are consistent with the dietary guidelines for Americans.
We are going to take a look at the different types of foods Hanna has for her friends and separate them into the food groups:
Meats : chicken and fish
Dairy : milk and cheese
Breads : spaghetti, brown rice, crackers, mixed nuts, dinner rolls
Fruits : mixed fruit, peaches
Vegetables : spaghetti sauce, lettuce
Fats, Oils & Sweets
Milk, Yogurt & Cheese
Meat, Poultry, Eggs
Vegetable Group Fruit Group
Bread, Cereal, Rice and Pasta Group
Notice that 5!, which is, 5 4 3 2 1, can be cancelled from 9!, 9! so could actually be written as 5!
× × × ×
× × × ×
! This now brings us to the formula,. The represents the number !( - )! of things that are available, and the represents the number of things you are choosing.
n r
n C n r n r r
=
Example 3 : A pizza parlor offers a selection of 8 different toppings. In how many ways can a pizza be made with 3 toppings?
8 represents n , the number of total toppings
3 represents r , the number of toppings you are choosing.
Replace these numbers in the formula and solve.
8 3
C =
8 3
C =
8 3
C
× × ×
8 3
C
× ×
× ×
8 C 3^ =^56
Sometimes you will need to find combinations of more than one thing at a time. In this case, multiply the combinations together to find the total amount of combinations.
There are 56 combinations of choosing 3 toppings from 8 selections.
Let’s go back to the pizza example and add different sizes to the list of choices.
Follow the example below.
Example 4 : A pizza parlor offers a selection of 8 different toppings and 3 different sizes. In how many ways can a pizza be ordered with the following selections: 2 sizes and 4 toppings?
3 2 8 4
3 sizes 8 toppings
choosing 2
choosing 4
C C
×
×
= ×
× × × × × ×
= ×
× × × ×
× ×
= ×
= ×
Thus, there are 210 combinations of 2 sizes and 4 toppings.
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