I am trying to find a closed form for the following summation:
$$\sum_{i=0}^k \binom{n}{i}\binom{m}{k-i}$$
, expecting a binomial coefficient without any summation notation as the answer.
I have broken down this summation as follows:
$$\sum_{i=0}^k \binom{n}{i}\binom{m}{k-1} = \binom{n}{0}\binom{m}{k}+\binom{n}{1}\binom{m}{k-1}+\binom{n}{2}\binom{m}{k-2}+...+\binom{n}{k}\binom{m}{0}$$
However, I am stuck on how to solve this after breaking it up like that. Someone had suggested to me that I should define the following terms and then take the product of the two expressions: $$(x+1)^n=\binom{n}{0}+\binom{n}{1}x+...+\binom{n}{n}x^n$$ $$(x+1)^m=\binom{m}{0}x^m+\binom{m}{1}x^{m-1}+...+\binom{m}{m}x^0$$
I am confused on how the above two terms have been derived and I am also confused on how these two terms can be used to find the closed form.
Can someone provide assistance?