The number of integers greater than 6000 that can be formed, using...

$4$ digit and $5$ digit numbers are possible.

$4$ digit numbers:

$\begin{cases}6 / 7 / 8 & & & \\ \overset{¯}{↑} & \overset{¯}{↑} & \overset{¯}{↑} & \overset{¯}{↑} \\ 3 & 4 & 3 & 2\end{cases} Total numbers possible = \text{3 . 4 . 3 . 2} = \text{72}$

$5$ digit numbers

$\begin{cases}\overset{¯}{↑} & \overset{¯}{↑} & \overset{¯}{↑} & \overset{¯}{↑} & \overset{¯}{↑} \\ 5 & 4 & 3 & 2 & 1\end{cases} Total numbers possible = \text{5 . 4 . 3 . 2 . 1} = \text{120}$

$∴ \text{Total numbers} = 72 + 120 = 192$