## Task 10/68

In chess, rooks can make moves of any length (as long as no square is occupied) in horizontal and vertical directions.

In how many ways can a rook be moved from one corner square to the diagonally opposite corner square, if no backward moves are allowed?
Without loss of generality, we assume that the starting field is the bottom left, and the target field is the top right.

The transition to the right-adjacent field is symbolized by $r$, and the transition to the above-adjacent field by $o$. Such moves are called "elementary moves". Any allowed sequence of moves, for example,

$$
3 r \ldots 2 o \ldots r \ldots 4 o \ldots r \ldots o \ldots 2 r
$$

can be broken down into a sequence of 14 elementary moves: rrrooroooororr.

Thus, we have a permutation of 14 elements rrrrrrrooooooo, where 7 elements are identical. The number of permutations in this case is

$$
P(14)=\frac{14!}{7!\cdot 7!}=3432
$$

The tower can reach its goal in 3432 different ways.