Counting Principles

Counting problems can be very simple and they can be quite challenging. Working on a challenging problems can be fun, but for now, I would like to introduce five basic concepts to help you get started with basic problems. Whenever you get stuck on a counting problem, try making up a smaller scale example of the same type problem. With the small scale example, you can list all possible answers and then figure out a more efficient way of calculating the answer that can be applied to the more challenging problem. Can’t figure out a smaller scale example? Try to get started on your problem, by beginning to list the possibilities. You may find that getting started listing the outcomes, helps you think about what you can do to get to your final count without finishing the list. Just do something more than random guessing to get your thought process started.

The five principles that you are expected to recognize when you see a problem are:

  1. Multiplication Principle
  2. Permutations
  3. Combinations
  4. Addition Principle
  5. Binomial Theorem
1.) Multiplication Principlie

My youngest daughter who is not fashion conscious is willing to wear any pants with any t-shirt, and any sweatshirt or sweater. If she has 5 pairs of pants, 10 t-shirts, and 3 sweaters/sweatshirts, how many outfits does she have if we also include the option of not wearing a sweater or sweatshirt?

When we start the list using the labels that I put on the articles of clothing we get:

Obviously, we could continue this process, but I would rather just look at what is happening. With only 1 of the pairs of pants, 3 of the t-shirts, and the 4 choices of sweaters or no sweater, we are already up to 12 possibilities. How do we get all of the possible answers?  Click here for the answer.

2.) Permutations

a.) I have 4 kids: Allie, Joey, Ken, and Sam and I want to spend some special one-on-one time with each of them this month. If I plan to give each kid one whole weekend of my time during the next 4 weekends, how many different orders can I do this in?
List all the possible orders using each childs first initial and then figure how you calculate the number. click here.

b.) In a class of 23 students, 4 are selected to be on the student council, where one is president, one is vice-president, one is treasure, and one is secretary. How many ways can the student council officers be selected? Click here.

3.) Combinations

Now suppose we are selecting 4 students out of 23 to represent us in a math contest. Now the order doesn't matter, so we could take our prior answer and reduce it by dividing by the number of arrangements possible. Click here.

4.) Addition Principle

Sometimes, you have to break a problem up into different cases and add the number of ways to satisfy each case. Suppose in a group of 7 students, we need at least 5 to go on a field trip. Let's calculate the number of ways to take students on the field trip.
How many ways can we take 5 students on the field trip?
How many ways can we take 6 students on the field trip?
How many ways can we take 7 students on the field trip?
How many ways can we take at least 5 students on the field trip?
5.) Binomial Theorem

Now suppose we have the same seven students and each one has a choice as to whether or not they go on the field trip. In the last example we already discussed the number of ways to take 5, 6, or 7 of the students. This easily leads to the number of ways to take 0, 1, or 2 students since taking 5 students is the same as leaving out 2 students which is the same as taking 2 students. Likewise, taking 6 students is the same as taking 1 student, and taking all seven students is the same as leaving out all of the students. Using this idea all we would need to calculate is taking 3 students which is the same as taking 4 students. Then we could add all of the answers. Write down the numbers with a space between them for taking 0 students, then 1 student, then 2 etc. and then click 'sumbit'.

We wouldn't want to have to use combinations and the addition principle every time we have a problem of this type, since that could be quite time consuming. An easier way to think about this would be to think in terms of the choices each student has. Each student has two choices. They can go on the field trip or not go. Using the multiplication principle that means for student one we have two choices, student two has two choices, etc. How do we calculate the answer using this idea?
Notice this is the same as adding the coefficients of the n=7 row of Pascal's triangle.
Counting Problems Involving More Than One Idea
Many problems that you will run across will involve several of the ideas listed above. The following example uses the multiplication principle, addition principle, and combinations. Like any application problem, it takes practice to get good at these and there can always be a problem that gives and expert a challenge. That is what makes them fun. They are like puzzles to be solved. Remember if you get stuck, try to begin listing the possibilities and/or make a small scale example.
Example:
A coed academic team needs to have at least one girl and at least one boy representing the team in any one round. The team at one particular school has 5 girls and 4 boys on it. How many ways can the coach assign players to a round if they must have 4 students playing at a time? Click here.