I tried to solve $\displaystyle \sum_{j=0}^{n} \sum_{i=j}^{n} \binom{n}{i} \binom{i}{j}$. Unfortunately, I cannot succeed in doing so. It comes with a hint that says: "Interchange the sums and first calculate $\displaystyle \sum_{j=0}^{i} \binom{i}{j}$ ". I do know how to solve this, but not how to come back to the original summation afterwards, since splitting up $\displaystyle \sum_{j=0}^{n}$ into $\displaystyle \sum_{j=0}^{i} + \sum_{j=i+1}^{n}$ results in an expression like $\displaystyle \sum_{i=j}^{n} \sum_{j=i+1}^{n}$ which I cannot make sense of.
Help would be appreciated.