A cute combinatorial identity

Just wanted to record a cute combinatorial argument. Prove that      \sum\limits_{i=0}^n {n\choose i} {i\choose m}= {n\choose m} 2^{n-m}

I read this formula in the article “Pascal functions” in “The American Mathematical Monthly”.

They prove it using the new machinery of Pascal functions that they develop in the article. I tried proving it using an algebraic argument, but couldn’t find the right formulation. I could finally prove it using a combinatorial argument, which I have explained below:

The left hand side counts the number of ways of first selecting i items from n items, and then selecting m items from those already-selected i items. Obviously, for i<m, this does not make sense. Hence, {i\choose m}=0 in those cases. For i\geq m, the above interpretation is valid.

We can interpret the right hand side as the number of ways of selecting m items from n items, and then going to each of the left-over items (there are n-m of them) and deciding if they had been pre-selected or not. Here pre-selection implies being selected amongst the i items, out of which the final m items have been chose.

Hence, the right hand side equals the left hand side.

Published by ayushkhaitan3437

Hello! My name is Ayush Khaitan, and I'm a graduate student in Mathematics. I am always excited about talking to people about their research. Please please set up a meeting with me if you feel that I might have an interesting perspective to offer- https://calendly.com/ayushkhaitan/meeting-with-ayush

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