19. From a bag containing $n-1$ white balls and $n-1$ black balls, all $2 n-2$ balls are drawn one by one, such that after each draw, the number of black balls remaining in the bag is not less than the number of white balls. Assuming balls of the same color are indistinguishable, find the number of all possible drawing sequences.
19. Represent the method of drawing balls by an ordered $(2n-1)$-tuple $\left(a_{0}, a_{1}, \cdots, a_{2 n-2}\right)$, where $a_{0}=0$. When a white ball is drawn at the $k$-th draw, $a_{k}=1$; when a black ball is drawn at the $k$-th draw, $a_{k}=-1$. Let $b_{k}=a_{1}+a_{2}+\cdots+a_{k}(k=0,1, \cdots, 2 n-2)$, then we have $b_{0}=0, b_{2 n-2}=0, b_{k} \geqslant 0$. By analogy with the previous problem, the number of ways to draw the balls is $C_{2 n-2}^{n-1}-C_{2 n-2}^{n}=\frac{1}{n} C_{2 n-2}^{n-1}$.