Chances are…

Counting:

Counting is a simple task that most of us had mastered by the age of 5 or 6. When the number of items that we are counting increases, counting becomes more difficult. There are several methods available to us to use when counting multiple items.

Additive Rule

The additive rule says sum the different items that you want to count.

Example: If I have 4 blue markers, 6 red markers, 8 yellow markers and 2 green markers, how many total markers do I have?

Solution:

4 markers + 6 markers + 8 markers + 2 markers = 20 total markers

Tree Diagram

A tree diagram gives a clear visual of what you are counting.

Example: Suppose we are choosing from the lunch menu at a certain restaurant. As appetizers the restaurant offers soup or salad. A club sandwich, cheeseburger or chicken strips are offered as your main dish. For dessert you can choose from a brownie with ice cream, a brownie without ice cream or silk pie. How many different meals can you order?

Solution:

 

There are 18 different meals from which you could choose.

 

Multiplication Rule

Using the previous example, we can present the menu in a different manner to show that it would be simple to use the multiplication rule for counting.

Appetizers

Soup

Salad

Main Dish

Club

Cheeseburger

Chicken Strips

Dessert

Brownie w/ ice cream

Brownie w/out ice cream

Silk Pie

 

Solution:

(2 appetizers) * (3 main dishes) * (3 desserts) = 2*3*3 = 18 choices

 

Try This:

Use both the tree diagram and the multiplication rule for solving this problem. Suppose you are buying a car. The car you are interested in comes in 3 colors (red, white and black). It is available with a standard or automatic transmission. This car offers a choice of 2 motors a 4-cylinder or a 6-cylinder. How many possibilities for this car do you have? If you want a white car, how many choices do you have?

 

Permutations:

Permutations are also used when counting items in a specific order. The formula used when finding permutations of r items taken from n items is

nPr = n!

(n-r)!

! means multiply n*(n-1)*(n-2)*…*(n-n+1).

Example: You have 6 numbered marbles in a bowl. How many permutations of 2 of these marbles are possible?

Solution: There are 6 choices for the first marble and 5 choices for the second marble. If you use the multiplication rule, you will find that (6 choices)*(5 choices)= 30 choices.

Or you can use the permutation formula to find that 6P2 = 6! = 6*5*4*3*2*1 = 30

(6-2)! 4*3*2*1

 

Combinations:

A combination is a choice of items in which the order is not important. From the previous example, for any 2 of the 6 marbles, there are 2! Ways to arrange them. You could choose the 1 marble first and the 2 marble second or the 2 marble first and the 1 marble second and so on. The number of combinations of r items chosen from n items is nCr = nPr = n!___ .

( r )! ( r )!(n-r)!

nPr is a way to find something when its order is important, since nCr does not take into account order, we must divide nPr by r!. Dividing by r! takes away the different arrangements of r items. The clearest way to explain this is by the coin toss problem.

Example: Toss two fair coins. The possible outcomes are (H,H), (H,T), (T,H), (T,T). There are 4 permutations because order is important. There are only 3 combinations because order is not important, the tosses (H,T) and (T,H) are considered 1 combination.