Chhandas and Binomial Theorem

Chhandas and binomial theorem? What connection can they possibly have? We can hear you asking that question. Well, read along to know the answer.

You know that chhandas uses 2 kinds of syllables - long and short. So does a binomial theorem, which uses 2 variables, say 'a' and 'b'.

You might have seen in the table that for 3 syllables, eight combinations have been used. 

Take a look at the eight gaNaas.  We have, according to the table,

           3 long syllables appearing once ( ma ),
           2 long and 1 short syllable appearing thrice ( bha, ja, sa ),
           1 long and 2 short syllables appearing thrice ( ya, ra, ta ),
           3 short syllables appearing once ( na ).

In fact, this is the arrangement for a class of metre which has 3 syllables per metre. 

Now consider this expression : (a + b)³

This can be expanded as : (a + b)³ = 1.a³ + 3.a²b + 3.ab² + 1.b³
If we assume that 'a' represents a long syllable, and 'b' a short one, we have :

    a³ corresponding to ma scheme,
    a²b term corresponding to the schemes bha, ja, sa,
    ab² term corresponding to the schemes ya, ra, ta, and
    b³ corresponding to na scheme.

As you can see, the binomial expansion of (a + b)³ corresponds to the layout of a 3 - syllable per metre class.

For a metre with 4 syllables, we have the following :

       1 metre with all 4 syllables long,
       4 metres with 3 long and 1 short syllable,
       6 metres with 2 long and 2 short syllables,
       4 metres with 1 long and 3 short syllables,
       1 metre with all 4 syllables short.

That is, there will be 16 ( = 1 + 4 + 6 + 4 + 1) different combinations for this class. This corresponds to the expansion :

         (a + b)⁴ = 1.a⁴ + 4.a³b + 6.a²b² + 4.ab³ + 1.b⁴

So you see how nicely binomial theorem fits in? Or rather, how the layout of chhandas supports the theory of binomial expansion?

This method was given in 200 B.C. by Pingala, the author of chhandaH-sootra (this is the book that defines the various metres, and their syllable patterns). This technique of finding the number of variations of sound was practiced by him and other composers, to detect the quality as well as shortcomings of the metres.

In the 10th century A.D., a method was developed by a scholar named  Halaayudha that would easily find the coefficients of the different variations in the expansion of (a + b)ⁿ_. This method, known as meruprastaara, is given below :

Well, this needs no explanation! This is the famous Pascal's triangle.Only, it was used by the Indians 4 centuries earlier than it appeared anywhere else in the world! Pascal arrived at this triangle only in the 16th century.

This, then, brings us to the end of the little journey we had undertaken to explore the world of metres and rhymes. Hope you have enjoyed it. This is but the tip of the iceberg, and there are many more such 'goodies' in the Sanskrit literature. It only requires you to get interested!

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