Proof of $\sum_{k=1}^n \binom{n}{k} \binom{n}{n+1-k} = \binom{2n}{n+1}$ via induction I'm having trouble to prove the following formula using Induction on $n \in \mathbb{N}$:
$$\sum_{k=1}^n \binom{n}{k} \binom{n}{n+1-k} = \binom{2n}{n+1}.$$
I've tried all the usual identities, but they seem to lead nowhere. Is there any trick to this, or is it just not possible to prove this using induction?
I'm thankful for any tip or advice on how to approach this :)
 A: Select $n+1$ unit squares from a $2\times n$ rectangle. What do you see?
A: As already noted yours is just a Vandermonde convolution, which is normally proved
by taking the product of two binomials.
However, if we limit to consider the general Vandermonde convolution with non-negative integer parameters, 
except at most one of the upper parametr which may also be real, i.e.
$$
\sum\limits_{\left( {0\, \le } \right)\,k\,\left( { \le \,m} \right)} {\left( \matrix{
  m \cr 
  k \cr}  \right)\left( \matrix{
  r \cr 
  n - k \cr}  \right)}  = \left( \matrix{
  r + m \cr 
  n \cr}  \right)\quad \left| \matrix{
  \;0 \le m,n \in Z \hfill \cr 
  \;r \in R \hfill \cr}  \right.
$$
then the convolution can be obtained as the $m$ times iteration of the basic
recursive identity
$$
\eqalign{
  & \sum\limits_{\left( {0\, \le } \right)\,k\,\left( { \le \,m} \right)} {\left( \matrix{
  m \cr 
  k \cr}  \right)\left( \matrix{
  r \cr 
  n - k \cr}  \right)}  = \left( \matrix{
  r + m \cr 
  n \cr}  \right)\quad \left| \matrix{
  \;0 \le m,n \in Z \hfill \cr 
  \;r \in R \hfill \cr}  \right.  \cr 
  & \left( \matrix{
  r + m \cr 
  n \cr}  \right) = \left( \matrix{
  r + m - 1 \cr 
  n \cr}  \right) + \left( \matrix{
  r + m - 1 \cr 
  n - 1 \cr}  \right) = \left( \matrix{
  1 \cr 
  0 \cr}  \right)\left( \matrix{
  r + m - 1 \cr 
  n \cr}  \right) + \left( \matrix{
  1 \cr 
  1 \cr}  \right)\left( \matrix{
  r + m - 1 \cr 
  n - 1 \cr}  \right) =   \cr 
  &  = \left( \matrix{
  r + m - 2 \cr 
  n \cr}  \right) + \left( \matrix{
  r + m - 2 \cr 
  n - 1 \cr}  \right) + \left( \matrix{
  r + m - 2 \cr 
  n - 1 \cr}  \right) + \left( \matrix{
  r + m - 2 \cr 
  n - 2 \cr}  \right) = \sum\limits_{\left( {0\, \le } \right)\,k\,\left( { \le \,2} \right)} {\left( \matrix{
  2 \cr 
  k \cr}  \right)\left( \matrix{
  r + m - 2 \cr 
  n - k \cr}  \right)}  =   \cr 
  &  = \quad  \cdots  = \sum\limits_{\left( {0\, \le } \right)\,k\,\left( { \le \,m} \right)} {\left( \matrix{
  m \cr 
  k \cr}  \right)\left( \matrix{
  r \cr 
  n - k \cr}  \right)}  \cr} 
$$
Now, the basic recursive identity can be taken as just the definition of the binomial ,
or if the binomial is defined in other ways it can be easily demonstrated also by induction.
And once you have demonstrated the basic recursion, its iteration leads you to demonstrate 
the Vandermonde convolution.
