Just a quick pointer: you could probably get away with skipping to the last 2 paragraphs, as the rest covers my motivation for answering this question.
Over this coming holidays I'm hoping to spend a little time reading through Spivak's Calculus, and answering some of the more difficult/interesting looking questions. While I've learnt most of the material before, the calculus courses in first year at my Uni were all taught primarily for engineers, and I feel I have a very 'handwavy' understanding of ideas like continuity, Riemann sums, etc., etc.
Anyway, I ran into this question in chapter 2 of the book last night:
Prove that:
$\sum_{k=0}^{l} \binom{n}{k}\binom{m}{l-k}=\binom{n+m}{l} $
Spivak recommended considering the binomial expansion of $(1+x)^a(1+x)^b$ and after a minute or so I saw the 'reason' why this statement is true. Briefly, we need to consider the coefficient of $x^l$ in the binomial expansion suggested above. Expanding out $(1+x)^{n+m}$ with the binomial theorem gives the coefficient $\binom{n+m}{l}$, the RHS of the expression above. On the other hand, if we expand out $(1+x)^n$ and $(1+x)^m$ separately and multiply them together, we're going to have to add up all of the different cross-multiplications that give an l'th order term: there's $\binom{n}{0}x^0\times \binom{m}{l}x^l$,$\binom{n}{1}x^1\times \binom{m}{l-1}x^{l-1}$, etc. And if we add these up we get the expression on the LHS of the equality above. These 2 expressions for the coefficient of $x^l$ must be equal, so therefore the equality above must hold.
Q.E.D.
Done.
Cool.
However, I'm still very uncomfortable with leaving the question with just this answer. The entire reason I'm reading through Spivak is to try to learn some mathematical rigour, and yet, while I'm sure this argument is valid, I have a feeling many analysts would want a little more... ummm...something :)
And this where I'm stuck. On one end of the scale, we could expect all proofs to be entirely based on axioms or previously proven theorems - Euclid style. At the other end we get arguments like this one above, or like Newton's intuitionistic view of a limit. My question is simply 'when should I be satisfied that a proof is a proof?' How much rigour is sufficient? Indeed, how much rigour is used in proofs in professional mathematics (I'd gather that's quite field-dependent)?
Thank you very much for your thoughts