The number of four-digit numbers strictly greater than $4321$ that can be formed using the digits $0,1,2,3,4,5$ (repetition of digits is allowed) is:

The correct option is A $310$
The numbers greater than $4321$ must have $4$ or $5$ in thousands place.

Case $1$: Let us fix first place as $4$
$4\_\_\_\_\_\_$
The second place can be filled in $2$ ways ($4,5$), third and fourth place can be filled in $6$ ways.
Number of such four-digit numbers $=2\times 6\times 6=72$

Case $2$: Let us fix first place as $4$ and second place as $3$
$43\_\_\_\_$
The third place can be filled in $3$ ways ($3,4,5$) and the fourth place can be filled in $6$ ways.
Number of such four-digit numbers $=3\times 6=18$

Case $3$: Let us fix first place as $4$, second place as $3$ and third place as $2$
$432\_\_$
The fourth place can be filled in $4$ ways ($2,3,4,5$)
Number of such four-digit numbers $=4$

Case $4$: Let us fix first place as $5$
$5\_\_\_\_\_\_$
Number of such four-digit numbers $=6\times 6\times 6=216$

So, total number $=72+18+4+216=310$