The Counting Principle, Pascal’s Triangle, and Powers of 2

This post shows how we can help students make connections among counting principle, the Pascal’s triangle, and powers of 2. I have tried this lesson in an in-service training program but I’ve yet to test it with students in high school. The lesson uses the strategy Teaching thru Problem Solving.

A piece of knowledge is powerful to the extent to which it is connected to other piece of knowledge. The more connections there are, the more powerful it becomes. Mathematics teaching therefore should always aim to help students make connections among the different concepts of mathematics. You may want to read  my article about   understanding as making connections.

The Problem: Trace the paths that will spell “MATHEMATICS” starting from the letter M on top moving only downwards, either to the immediate letter to its right or to the immediate letter to its left. How many different paths are there in all?

After a few minutes and the class is seem getting nowhere you may suggest to students to try simpler case first  like trying the word MATH. Trying simpler case is a good problem solving strategy and habit students need to learn.

Solution 1

Suppose we  spell the word “MATH” only. From M we can move downwards and may either choose the A at the left or the A at the right. Having chosen an A we can either choose the T down left or the T down right. And having chosen one we can either choose the H down right or the H down left. Each time we only have two choices. Thus, the number of ways of tracing the word “MATH” in the above figure is

2·2·2 =2³=8

Using the same line of thinking, the total number of paths which spells “MATHEMATICS” is

 2·2·2·2·2·2·2·2·2·2=2¹⁰=1024.

Solution 2

Notice the number of arrows that converges to a particular letter. It tells the number of paths that pass through it. Thus, to count the number of ways of tracing the word “MATH” we only have to add the total number of arrows that point to the H’s. There are

 1 + 3 + 3 + 1 = 8.

Count the number of arrows converging to each letter in MATHEMATICS . You will generate the triangular array of numbers below.

The number of arrows converging to S is

1 + 10 + 45 + 120 + 210 + 252 + 210 + 120 + 45 + 10 + 1 = 1024 or  2¹⁰.

The solutions showed two important principles of counting.

The Multiplication Principle. If one task can be done in m ways and then another task can be done in n ways, the pair of tasks, first one and then the other, can be performed in

m • n ways.

 The Addition Principle. If one task can be done in m ways and another task in n ways, then one task or the other can be done in

m + n ways.

Anyone who wants to understand permutations, combinations and anything that involves counting should first understand these principles.

The triangular array of numbers generated above is one of the most influential number patterns in the history of mathematics. It is called Pascal’s triangle after the renowned French mathematician Blaise Pascal (1623-1662) who discovered it. The triangle is also called Yang Hui’s triangle in China as the Chinese mathematician Yang Hui discovered it much earlier in 1261. The same triangle was also in the book “Precious Mirror of the Four Elements” by another Chinese mathematician Chu-Shih-Chieh in 1303.

The Pascal triangle yields interesting patterns and relationships. Some of the obvious ones are:

1. To generate the next row, you will have to add the two numbers above it.
2. Another striking property of this array of numbers is its symmetry. Note the numbers on both sides of the middle number in each row.
3. The sum of the numbers in each row can be expressed in powers of two.

Recommended readings on combinatorics:

1. Mathematics of Choice: Or, How to Count Without Counting (New Mathematical Library)
2. Counting: The Art of Enumerative Combinatorics (Undergraduate Texts in Mathematics) 
3. Introductory Combinatorics (5th Edition)