<s>[INST] Proving $\binom{n}{m}+2\binom{n-1}{m}+......+(n-m+1)\binom{m}{m} = \binom{n+2}{m+2}$ 
For $m,n\in\mathbb{N},\;n\geq m$, prove the following:
$$
\tag{i}\binom{n}{m}+\binom{n-1}{m}+\binom{n-2}{m}+......+\binom{m}{m} = \binom{n+1}{m+1}
$$
  $$
\tag{ii}\binom{n}{m}+2\binom{n-1}{m}+3\binom{n-2}{m}+......+(n-m+1)\binom{m}{m} = \binom{n+2}{m+2}
$$

My Attempt:
For $(\mathrm{i})$, We can write $\binom{n}{m}$ as the coefficient of $x^m$ in $(1+x)^n$. Thus we can also write $\binom{n-1}{m}$ as the coefficient of $x^m$ in $(1+x)^{n-1}$ and $\binom{n-2}{m}$ as the coefficient of $x^m$ in $(1+x)^{n-2}$. So we have to find the coefficient of $x^m$ in
$$
(1+x)^n+(1+x)^{n-1}+(1+x)^{n-2}+........+(1+x)^{m}
$$
Using the formula for the sum of a geometric progression, this sum equals
$$\frac{(1+x)^{n+1}-(1+x)^m}{x}$$
So we now need to find the coefficient of $x^m$ in
$$
\frac{(1+x)^{n+1}-(1+x)^m}{x}
$$
or, equivalently, we need to find the coefficient of $x^{m+1}$ in
$$
(1+x)^{n+1}-(1+x)^m = \binom{n+1}{m+1}
$$
We can use a similar method to solve $(\mathrm{ii})$. Can these questions be solved using combinatorial methods instead?
 [/INST] Maybe a bit late now but here is my answer:
Consider counting the number of possible bit-strings of length $n+i$:
$$ \underbrace{111...\overbrace{\textbf{1}}^{i^{\text{th}}\ 1}...111}_{m+i\ 1\text{s}}\ 000...000 $$
So, here $n$ is the number of digits to the right of the $i^\text{th}\ 1$. 
The number of bit strings is given simply by $\binom{n+i}{m+i}$ where $m$ is the number of $1$s to the right of the $i^{\text{th}}\ 1$.
We may also count bit strings for each possible position of the $i^{\text{th}}\ 1$ (i.e. position $i$, position $i+1$ up to position $i+n-m$) and sum these to give the same result.
When the $i^{\text{th}}\ 1$ is in a position $i+r$ there are $i+r-1$ digits to the left including $i-1$ $1$s and $n-r$ digits to the right including $m$ $1$s.
So if we sum over $r$:
$$\dbinom{n+i}{m+i} = \sum_{r=0}^{n-m} \dbinom{i+r-1}{i-1}\dbinom{n-r}{m} $$
Your two examples are special cases of this identity.
If we put $i=1$ and $i=2$ we have the two required results:
$$\dbinom{n+1}{m+1} = \sum_{r=0}^{n-m} \dbinom{r}{0}\dbinom{n-r}{m} = \sum_{r=0}^{n-m} \dbinom{n-r}{m} = \dbinom{n}{m} + \dbinom{n-1}{m} + ... + \dbinom{m}{m} $$
and
$$\dbinom{n+2}{m+2} = \sum_{r=0}^{n-m} \dbinom{r+1}{1}\dbinom{n-r}{m} = \sum_{r=0}^{n-m} \left(r+1\right)\dbinom{n-r}{m} = 1\dbinom{n}{m} + 2\dbinom{n-1}{m} + ... + \left(n-m+1\right)\dbinom{m}{m} $$
</s>