248 Part 4: The State of the World
Counting Methods ..................................................................................................................
You know how to count, of course. If you had a bag of marbles, some red and some blue, and
you wanted to know the chance that the first marble you pulled out (without looking) was a
blue marble, the first thing you’d want to know is how many red marbles and how many blue
marbles are in the bag. Unless it was a gigantic bag of marbles, you could answer that question by
counting.
On the other hand, suppose you were going to buy a lottery ticket that asked you to pick five
numbers from a group of 50, and you had to get all five correct to win. Your chance of winning
will depend on how many ways there are to pick five numbers, and that counting problem is
more complicated.
When you start to pick your number for the lottery ticket, you have 50 numbers to choose
from. After you pick your first number, there are 49 left to choose from for the second number,
then 48 for the third, 47 for the fourth, and 46 for the fifth. (That’s assuming you have to pick
five different numbers. If you’re allowed to repeat, you have 50 choices each time.) And there’s
another question, too. Do you have to have the five numbers in just the right order? Is 12345 the
same as 54321? Or are those different?
Big counting jobs raise many questions, but there are some rules that will help. Let’s start with
the most essential rule.Basic Counting Principle
The fundamental rule for quick counting is called the basic counting principle. It gives you a fast
way of counting up the possibilities by multiplying the number of choices. Suppose you’re getting
dressed and you need to choose jeans or khakis and then pick a shirt from the six t-shirts in the
drawer. You could wear any one of the six t-shirts with jeans, or any one of the six t-shirts with
khakis. That’s 12 possible outfits: 2 choices for pants times 6 choices for shirts.
If you have to do something that requires several choices, and you create a slot for each choice
that needs to be made and fill each slot with the number of options for each choice, multiplying
those numbers will tell you how many different possibilities you have.
If you have 5 shirts, 4 pairs of slacks, and 3 pairs of shoes, and you’re comfortable mixing and
matching any of them, how many different outfits can you make? Using the basic counting
principle, create a slot for shirts, a slot for slacks, and a slot for shoes:
shirts slacks shoes.
Then fill in the number of each that you have and multiply.
54 60
shirts slacks shoes outfits3 353