I'm working through Richard Hamming's "Methods of Mathematics Applied to Calculus, Probability, and Statistics" on my own.
I'm struggling with his proof of the binomial theorem, as summarized below.
He starts by establishing the fundamental relation between the coefficient of the kth term in the nth row of Pascal's triangle:
which basically just states that within Pascal's triangle each number is the sum of the two numbers in the row above.
He then claims that C(n,k) is also equal to:
$$n! \over k!(n-k)!$$
Let's call this "The Formula."
Next, Hamming confirms that The Formula holds for the case where n=1 with the sub-cases k=0 and k=1. This serves as the basis for the induction reasoning.
Then, after assuming The Formula holds for the case m-1, Hamming shows that this implies The Formula holds for m.
So far so good!
Here's where he loses me. I'm going to paste the actual paragraph:
(Note: formula 2.4-5 is The Formula.)
First off, I think there's a typo. He says we verified the induction for the case k=1, but I think he meant n=1?
Related, he says when we moved from the (m-1)th case to the mth case we only did so for k=1,2,...,(m-1), thus omitting k=0 and k=m. I can't see why/how this is so. To form the basis of the induction, we started with n=1 and tested k=0 and k=1.
Can someone weigh in and show me what I'm missing? Much, MUCH, appreciated!