General: Sum of binomial coefficients

[PDF]Sum of products of binomial coefficients. A gemeral appraoch

During my secondary school days I used to read books related to math and physics for no reason. If I got any book anywhere; I would read it no matter whether I understood it or not. I used to spend hours imagining what would the fact mean in reality; because I wouldn’t know what was the real fact hidden in it. I used to try to solve problems given and used to get happy. My classmate would probably call my habit of reading such books without getting anything useful(?) out of it as a show off but it meant a real pleasure for me.

I had one of my brothers studying in high school. I got a book on entitled Intermediate algebra. In the book I read (not quite studied) about permutation; combination; binomial theorem; complex numbers etc which was not a matter of seconday school course in mathematics. I would never get full out of it nevertheless I never gave up reading those books.

From those days I was so much fascinated by the elegance of binomial theorem. Getting to learn that Newton formulated binomial theorem made me so much amazed what a individual genius can do in different field (I had already read about Newton’s laws and those fascinating stories of falling apple on Newton’s head before). I found binomial theorem so much general. I loved general proofs or general approach to any problem.

I have a story related to my fascination towards general approach. As said earlier my reading of books related to physics without knowing the real truth had a impression on me. On  any book somewhere I had read scientist were searching for a “Unified theory”. This lead me to think what would a unified theory do. Somewhere earlier. I had found that pythagorus theorem was a special case of cosine law; my guess about unified theory was based upon this fact and thought probably scientist were searching for such a theory that every other theory would be the special case of the unified theory (I still don’t know about it) as Pythagorus theorem would be a special case of coseine law and cosine law must also be special case of some other law for the special case of triangle and the law would another law for quadrilateral. This made a deep impression on me that we have to go towards finding genereal approach on any problems and special cases of these would yield another useful existing or non existing theory.

During those days my fascination towards general approach was reflected in my Programming classes too ( I learnt QBasic programming language those days in fact I still do). To do a simple task like printing a sequence of odd numbers I would use lot of variables and lots of codes that it would be general. Mere change of a variable would lead to another useful sequence out of that same coding.

Following is a general approach of finding the sum of product of binomial coefficients.

I love general approach to any problem. Hope you like it. Comments and suggestions are always welcome.

Prakash Gautam

pranphy@hotmail.com

Bhaktapur.

[PDF]Sum of products of binomial coefficients. A gemeral appraoch

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