  Using properties of shuffles of copulas and tools from combinatorics we solve
the open question about the exact region $\Omega$ determined by all possible
values of Kendall's $\tau$ and Spearman's $\rho$. In particular, we prove that
the well-known inequality established by Durbin and Stuart in 1951 is only
sharp on a countable set with sole accumulation point $(-1,-1)$, give a simple
analytic characterization of $\Omega$ in terms of a continuous, strictly
increasing piecewise concave function, and show that $\Omega$ is compact and
simply connected but not convex. The results also show that for each $(x,y)\in
\Omega$ there are mutually completely dependent random variables whose $\tau$
and $\rho$ values coincide with $x$ and $y$ respectively.
