The problem asks to find the total number of paths for a rook on chessboard to move from the south west corner to northeast corner of the board by moving eastward and northward only which consist of four eastward moves and three northward moves.(By an eastward move we mean a certain number of consecutive eastward steps. A northward move is defined similarly. )
Now I get that there are a total of $\frac{14!}{7!7!}$ paths from southwest corner to northeast corner but I can't figure out how to calculate total paths which consist of four consecutive eastward steps and three consecutive northward steps.
Hint. Your problem is equivalent to counting the number of positive integer solutions of the following system $$\begin{cases} x_1+x_2+x_3+x_4=7\\ y_1+y_2+y_3=7 \end{cases}$$ where $x_i\geq 1$ is the length of the $i$-th eastward move and $y_j\geq 1$ is the length of the $j$-th northward move. Then a path is obtained as $x_1$ steps eastward, $y_1$ steps northward, $x_2$ steps eastward, $y_2$ steps northward, $x_3$ steps eastward, $y_3$ steps northward, and $x_4$ steps eastward.