Weird Induction...? I was watching this video earlier and I couldnt figure out why the following step was possible. This is the original problem:
$\sum_{i = 0}^{n} \binom{n + i}{i} = \binom{2n + 1}{n + 1}$  
At one point, this somehow becomes:  
$\sum_{i = 0}^{n} \binom{n + i}{i} = \binom{n + k + 1}{k}$ for $0 \leq k \leq n$
Since it was a video, I couldnt really ask the person... I'm very stuck trying to figure this out...

Edit: the link has been added above. You can also find it here. The link should start the video at 4:44, where the relevant formula appears.
 A: I believe that you are either misinterpreting the video, the video is wrong, or something else is going on because, if
$\sum_{i = 0}^{n} \binom{n + i}{i} = \binom{2n + 1}{n + 1}$
and
$\sum_{i = 0}^{n} \binom{n + i}{i} = \binom{n + k + 1}{k}$ for $0 \leq k \leq n$,
that would imply
$\sum_{i = 0}^{n} \binom{n + i}{i} = \binom{2n + 1}{n + 1} = \binom{n + k + 1}{k}$ for $0 \leq k \leq n$,
and, more specifically,
$\binom{2n + 1}{n + 1} = \binom{n + k + 1}{k}$ for $0 \leq k \leq n$.
However, I can show you a counterexample of the fourth statement when $n = 4$ and $k = 2$:
$\binom{2 * 4 + 1}{4 + 1} = \binom{4 + 2 + 1}{2}$ and $0 \leq 2 \leq 4$
$\binom{9}{5} = \binom{7}{2}$
$126 = 21$
The last statement is of course a contradiction, so the premise is false.

Edit: I suspect it may be the case that
$\sum_{i = 0}^{k} \binom{n + i}{i} = \binom{n + k + 1}{k}$ for $0 \leq k \leq n$,
but that's just a hunch. I may investigate further.

Edit:
First of all, the second equation you wrote in your question is not used in the video. It should actually be this:
$\sum_{i = 0}^{k} \binom{n + i}{i} = \binom{n + k + 1}{k}$ (Eq. 1)
Note the $k$ in the summation.
Second of all, when we look at the video, at 3:37 she writes
$\sum_{i = 0}^{k} \binom{n + i}{i} = \binom{n + k + i}{k}$,
but it should be Eq. 1 (from above). In the video she writes $n + k + i$ in the binomial coefficient when it should be $n + k + 1$.
She then proves that Eq. 1 is true using induction. This is useful for reasons that you may already understand, but if not, here's an explanation:
If Eq. 1 is proven true, then we can let $k=n$, then replace all the $k$'s with $n$'s. If we do that, then we get this:
$\sum_{i = 0}^{n} \binom{n + i}{i} = \binom{n + n + 1}{n}$.
Now, as she says at 9:19,
$\binom{A + B}{A} = \binom{A + B}{B}$.
In our case, $A=n$ and $B=n+1$, so we can write
$\binom{n + n + 1}{n} = \binom{n + n + 1}{n + 1}$,
and thus
$\sum_{i = 0}^{n} \binom{n + i}{i} = \binom{n + n + 1}{n} = \binom{n + n + 1}{n + 1}$,
which can also be expressed as
$\sum_{i = 0}^{n} \binom{n + i}{i} = \binom{2n + 1}{n}$.
So as long as Eq. 1 is proven true, we can prove the question statement. The video proves Eq. 1 is true using induction.
