# Basic counting rules

Many counting problems require one to list all possibilities, and count them. Sometimes drawing a tree helps to list all the possibilities. Many problems can only be dealt with with ad hoc methods. But there are two techniques or rules which are often employed, alone, or within more complicated methods. We shall illustrate these with a dessert buffet which has vanilla and butter pecan ice cream; and apple, cherry, and blueberry pie.
If you are going to have ice cream or pie for dessert, there are two choices (vanilla or butter pecan) if you have ice cream, and three choices (apple, cherry, or blueberry) if you have pie. Therefore there are 2 + 3 = 5 choices for dessert. If you are going to choose from one category or anther category, then you add the number of choices in each category. This is the addition rule for or.

SOme people believe in pie a la mode, which entails pie with ice cream on top, i.e., one must choose a flavor of pie and a flavor of ice cream. One can list the possibilities: apple with vanilla, apple with butter pecan, cherry with vanilla, cherry with butter pecan, blueberry with vanilla, blueberry with butter pecan; and count that there are six possibilities. Or one can note that one must first choose a flavor of pie and then choose a flavor of icecream: 3 × 2 = 6. When one must make a choice from one category and a choice from another category, one multiplies the numbers of choices in the two categories. This is the multiplication rule for and.
Lest we think that counting problems are all trivial, consider a larger dessert buffetwhich in addition to vanilla and butter pecan ice cream and apple, cheery, and blueberry pie, also has chocolate, lemon, raspberry, and strawberry cheesecake. The buffet is supervised by a dietician who insists that pie be eaten with ice cream, but cheese cake be eaten alone. Hence you must choose cheesecake or pie a la mode, which can be rewritten with parentheses as cheesecake or (pie and ice cream). from the above rules identifying or with + and and with ×, this becomes 4 + (3 × 2) = 10 posible desserts.

Exercise: John has three pairs of slacks, two turtleneck shirts, five shirts with collars, and four neckties. How many outfits does he have?

The multiplication rule extends to more than two choices. For example, how many different three number combinations are possible for a combination lock with the numbers 0 through 39? There are 40 choices for the first number, forty choices for the second number, and forty choices for the third number. Note that you must choose a first number and a second number and a third number. And means multiply, so there are 40 × 40 × 40 = 64,000 possible combinations.
How many license plates are possible if every license plate is three letters followed by three digits? There are 26 choices for each letter and 10 choices for each digit. 26 × 26 × 26 × 10 × 10 × 10 = 17,576,000 possible different license plates.

**Competencies:**If there are three tours to England and two tours to Ireland, how many different vacations are possible if a vacation consists of a tour of England or a tour of Ireland? If a vacation consists of a tour of England and a tour of Ireland?

If there are five tours of the continent in addition to the tours of the British Isles mentioned above, and a vacation consists of a tour of the continent, or a tour of England and a tour of Ireland, how many vacations are possible?
**Reflection:**

**Challenge:**How many integers have fewer than four digits? How can you obtain that answer from the methods discussed above?

May 2002

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