Given,

$x$ is chosen at random from the set $\left\{1,2.3,4\right\}$,

And $y$ is chosen at random from the set $\left\{5,6,7\right\}$,

Now given that xy will be even, then number of outcomes will be $\left\{\left(1,6\right),\left(2,5\right),\left(2,6\right),\left(2,7\right),\left(3,6\right),\left(4,5\right),\left(4,6\right),\left(4,7\right)\right\}\to 8 \text{outcomes}$

Now total sample space will be $4\times 3=12$,

Now probability will be $\frac{8}{12}=\frac{2}{3}$