Sum - can I just change the terms? I am looking at an exercise where this sum appears $\sum_{k=0}^{n}\binom{2n+1}{2n+1-k}$, and I saw in my textbook that it should be equal to $\sum_{u=n+1}^{2n+1}\binom{2n+1}{u}$.
I replaced at the first sum $2n+1-k$ with $u$, and I get this sum: $\sum_{u=2n+1}^{n+1}\binom{2n+1}{u}$. How did they find the result? Can I just change the terms $n+1$ and $2n+1$, because $n+1$ is smaller?
 A: It sometimes helps to write out the terms.
\begin{align*}
\sum_{k=0}^n\binom{2n+1}{2n+1-k} &= \binom{2n+1}{2n+1} + \binom{2n+1}{2n} + \dots + \binom{2n+1}{n+2} + \binom{2n+1}{n+1}\\
\\
\sum_{u=n+1}^{2n+1}\binom{2n+1}{u} &= \binom{2n+1}{n+1} + \binom{2n+1}{n+2} + \dots + \binom{2n+1}{2n} + \binom{2n+1}{2n+1}.
\end{align*}
So the two sums are the same, they just reverse the order of the summands. 
To see this via a change of index, let $u = 2n+1-(n-k)$ so that $u = n+1+k$; as $k$ goes from $0$ to $n$, $u$ goes from $n+1+0 = n+1$ to $n+1+n = 2n+1$. You can think of this as a combination of two index changes, first $w = n - k$ reverses the order of the terms, and then $u = 2n+1-w$ shifts the index. 
A: Apllying theorem of smmetry,we have that.
$$\sum_{k=0}^{n}\binom{2n+1}{2n+1-k}=\sum_{k=0}^{n}\binom{2n+1}{k}$$
and from 
$$\sum_{k=0}^{2n+1}\binom{2n+1}{k}=\sum_{k=0}^{n}\binom{2n+1}{k}+\sum_{k=n+1}^{2n+1}\binom{2n+1}{k}$$
follow that
$$\sum_{k=0}^{n}\binom{2n+1}{k}=\sum_{k=n+1}^{2n+1}\binom{2n+1}{k}$$
A: $\sum_{k=0}^{n}{{2n+1}\choose{2n+1-k}}=\sum_{k=0}^{n}{{2n+1}\choose{k}}=\sum_{k=n+1}^{2n+1}{{2n+1}\choose{k}}$
The first equality holds because it holds for each term. The second equality holds because the sum of the first half of the $2n+2$ coefficients equals the sum of the second half.
Clarification: I assume you know
$$
     {{2n+1}\choose{k}} =\frac{(2n+1)!}{k!(2n+1-k)!}=\frac{(2n+1)!}{(2n+1-k)!k!}= {{2n+1}\choose{2n+1-k}}.
$$
That's how you get the first equality above. The second equality, comes from the above pairing, but where the terms in the third sum are written in opposite order as in the second sum.
