"chess" problem Suppose that we have a cheespiece that can move either to the left or upwards. How many possibilities are there on a $n^2$ board to come from the bottom right square to the upper left square of the board.
I tried to show that this in fact is $2^{2n-1}$ but I don't get the induction right.

Consider the fact that no matter which path the piece takes, it must go left $n$ times and up $n$ times. Thus the number of all possible paths is all the possible permutations of the multiset $\{n \cdot \operatorname{up},n \cdot \operatorname{left}\}$ which has a closed form formula.
