The number of integral values of $k$, for which one root of the equation $2x^2-8x+k=0$ lies in the interval $\left(1,2\right)$ and its other root lies in the interval $\left(2,3\right)$, is : 

Let $f\left(x\right)=2x^2-8x+k=0$

$f\left(1\right)>0$

$f\left(2\right)<0$

$f\left(3\right)>0$

Now, $f\left(1\right)>0\Rightarrow 2-8+k>0\Rightarrow k>6 ......\left(1\right)$

$f\left(2\right)<0\Rightarrow 8-16+k<0\Rightarrow k<8 ......\left(2\right)$

$f\left(3\right)>0\Rightarrow 18-24+k>0\Rightarrow k>6 ......\left(3\right)$

From equations $\left(1\right), \left(2\right) \& \left(3\right)$,

$k\in (6,8)$

$\therefore$ Integral value of $k=7$

Therefore number of integral value of $k$ is one