  We report the first observation of $B^0 \to X(3872) (K^{+}\pi^{-})$ and
evidence for $B^+ \to X(3872) (K^{0}\pi^{+})$. The product of branching
fractions for the former decay mode is measured to be ${\cal B}(B^0 \to X(3872)
(K^+ \pi^-)) \times {\cal B}(X(3872) \to J/\psi \pi^+ \pi^-) = (7.9 \pm
1.3(\mbox{stat.})\pm 0.4(\mbox{syst.})) \times 10^{-6}$ and also find that
$B^{0}\to X(3872) K^{*}(892)^{0}$ does not dominate the $B^{0}\to
X(3872)K^{+}\pi^{-}$ decay mode in contrast to other charmonium states like
$\psi'$. The product of branching fractions for the latter decay mode is
measured to be ${\cal B}(B^+ \to X(3872) (K^0 \pi^+)) \times {\cal B}(X(3872)
\to J/\psi \pi^+ \pi^-) = (10.6 \pm 3.0(\mbox{stat.}) \pm 0.9(\mbox{syst.}))
\times 10^{-6}$. This study is based on the full and final data sample of
711~fb$^{-1}$ ($772\times 10^6 B\bar B$ pairs) collected at the $\Upsilon(4S)$
resonance with the Belle detector at the KEKB collider.
