**Abstract: **
By a geometrical treatment of the Bethe ansatz, an exact solution
for the totally asymmetric exclusion process is derived. An explicit
determinant expression is obtained for the non-stationary conditional
probability $Prob(x_1,...,x_P;t|x_1^0,...,x_P^0;0)$ of finding $P$
particles on sites $x_1,...,x_P$ at time $t$ provided they are on sites
$x_1^0,...,x_P^0$ at time $t=0$. Thus, a complete solution of the master
equation for a system of interacting particles is obtained.