Complex Counting

< 5.6 Simple Counting | Topic Index | 5.8 Cardinal &amp; Ordinal Numbers >

Though counting starts as a simple idea, in real-life it can get complicated requiring sophisticated methods of counting. We saw that even simple counting is actually a sophisticated skill consisting of 3 separate steps.

We will first see how the steps in simple counting themselves can be made more complex and sophisticated.

The sophistication of the second step can be varied to develop the ability to think visually & logically.

Firstly physical touching the items to be counted can be discouraged. This means the one-to-one correspondence has to be done mentally and the sequence remembered. The process can be started with a small number and then increased as the student masters the "mental" counting process.

A set of tokens can be spread randomly on a table and the student asked to find the total without touching the tokens.

Next step can be to count the "red" tokens from a set of "red & green" tokens spread randomly on the table.

Next step can be counting embedded shapes. For example, counting the number of squares in the shape given below.



The figure contains squares of 3 different sizes. To count correctly, a logical procedure would be helpful to avoid double counting. The different sized squares can be counted one after the other, for example 3 X 3 then 2 X 2 &amp; then 1 X 1.

While counting the 2 X 2 squares, a high level of visual discrimination would be required to count all possible squares at the same time avoiding double counting.

The result can be captured as a formula connected to the size of the square (here 3) and extended to square of any dimension.

From these basic ideas, the idea of counting has developed into the topics of permutations &amp; combinations &amp; combinatorics. Combinatorics is an important branch of modern mathematics.

The idea of permutations can be captured by this problem. Using numbers 1, 2 &amp; 3, how many different 3-digit numbers can be formed without repeating any of the above numbers?

According to Prof Manjul Bhargava, one of the early contributions to this science was the study of various &ldquo;verse meters&rdquo; in the hymns of the Vedas by Pingala in his Chandas Sutra written in 2nd century BC.

Combinatorics

The idea of complex counting led to a new branch of math called Combinatorics. In simple words, it is the math of counting things.

Complex counting also leads to the topic of permutations & combinations and probability.

< 5.6 Simple Counting | Topic Index | 5.8 Cardinal &amp; Ordinal Numbers >