Help Your Child Win the Race of Counting

Winners have something losers often fail to have. In education and mathematics, it's a solid background and clarity in how to manipulate basic concepts. This note focuses on the most basic concepts in counting, the Fundamental Counting Principles.

A father called and asked me to assist his son in preparing for his algebra 2 final. Before the actual meeting, the father emailed me what his son perceived were his difficulties: permutations, combinations, and probability. During our introductory conversation, it became clear what the boy's problems were.

First, the language he used to express mathematical concepts was imprecise. He called probability, possibility; and he confused combinations with permutations. Second, he couldn't recite the Fundamental Counting Principle. But to his credit, he was quite adept at manipulating his calculator to compute nCr : a combination of "n" things taken "r" at a time.

This note contains the thoughts and suggestions of that meeting. I ended the meeting after checking the student's mastery of the principles; by asking him to solve some counting problems. The meeting lasted about 130 minutes.

My first priority: Introduce him to the fundamental rules of counting.

The Fundamental Counting Principle is the stage upon which all counting is done. It must be memorized or students will not solve problems correctly.

My second priority: Impress upon him the gravity of imprecise language.

Mathematics is very unforgiving to users of imprecise language. Ideas must be clearly expressed verbally, and especially in writing. Slangs in speech and fragments in written expressions will soon doom anyone to the deep pit of confusion and trouble.

Fundamental Counting Principle (FCP)
If activity A can be done in m ways and activity B can be done in n ways then the total number of ways of doing activity A followed by activity B is mn.

EXAMPLE 1: John can swim the entire length of the school's pool and back using anyone of three different strokes. At the last swim practice, the coach instructed the swimmers that they can use any stroke they know to get to the other end. However, on the return trip, they may not use the same stroke. In how many ways can John obey the coach's instructions?

Since all counting takes place subject to the FCP, I must identify the activities; and the number of ways these activities can be done.

To obey the coach's rules, John must do two activities.

Activity A: Swim to the far side.
Activity B: Return to the start using a different stroke.

If John performs Activity A followed by Activity B, he will obeying the coach's instructions.

John can do Activity A in 3 ways.

John can do Activity B in 2 ways. (The coach's instructions forbid using the same stroke both ways.)

FCPsay the total number of ways to do activity A followed by activity B is 3•2=6 ways.

To gain clarity of how FCP works, the activities and the number of ways they can be done must be written.

When you write the details, it's easy to verify whether the activities give the desired action. In this case, it's obeying the coach's swimming rules.

Here's a jingle that aids memorizing the FCP.
(Use a steady clap rhythm.)

Activity A in m ways
Activity B in n ways
m times n, the number of ways for activities A and B.

The bolded words in the jingle occur on the beat (the clap).

The phrase " to do A and B " in the jingle represents the phrase " to do activity A followed by activity B ".

EXAMPLE 2: During his summer vacation at the family's beach home John never wears anything on his feet. He dresses himself by putting on a shirt and a pair of pants or a pair of shorts. He brought 5 shirts, 3 pairs of shorts and 8 pairs of pants to the beach house. In how many possible ways can John dress himself for an evening party?

Since this is a counting problem, I use FCP.
To dress himself, John must do two activities.
Now, identify the activities and the number of ways of ways they can be done. (Put in parenthesis).

Activity A: Put on shirt. (5)
Activity B: Put on pants. (11)

FCP says there are 55 ways for John to dress himself.

The Second Fundamental Counting Principle (FCP2)
If Activity A can be done in m ways and activity B can be done in n ways then the total number of ways of doing activity A or by activity B is m+n.

A Jingle help to memorize FCP2.

Activity A in m ways,
Activity B in n ways,
m plus n, the number of ways for activity A or B.

EXAMPLE 3: The northern branch of Highway 103B ends at its intersection with Highway 74. From that intersection, the eastern branch of Highway 74, has 5 towns. In the western direction, there are 20 towns. How many possible towns can John visit if he drives along Highway 74?

Since this is a counting problem, I use FCP2.
To drive along Highway 74, John must go east OR west from the intersection.

Activity A: Go to the east (5)
Activity A: Go to the west (20)

FCP says there are 25 possible towns John can visit.

Note the differences between the two counting principles. In the first case, both activities are done. In the second, one or the other is done (one does not follow the other.)

CHECK THE STUDENT:
A single card is drawn from a bridge deck. In how many ways can the selected card be a face card or a card with less than four symbols? Ask the student to tell which counting principle he would use. Ask why. Have him to compute the number of ways. ( 24 ways, FCP2 ) The ace has one symbol on it.

Two cards are selected from a bridge deck at the same time. In how many ways can one display the results, if the order of displaying the card does not matter? Which counting principle should be used? Compute the number of ways.
( 1326 ways, FCP )

Instruct the student to practice this lesson using the problems in his textbook.