How many $4-digit$ numbers greater than $2300$ can be formed with the digits $0,1,2,3,4,5$ and $6$, no digit being repeated in any number?

There are two cases, $\left(i\right)$First digit is greater than $2$

$\left(ii\right)$First digit is $2$

Case $:\left(i\right)$ First digit can be chosen from $3,4,5,6$ so $4$ possibilities,

then next $3$ digits can be selected from any $6$ digits.

$=4\times 6\times 5\times 4=480$

Case $:\left(ii\right)$ First digit is $2$, second digit starts from $3$, so it can be $3,4,5,6$ so $4$ possibilities.

Now if any one digit of third and fourth digit can be $0$ as no digit is repeated so it will be surely greater than $2300$. So last two digits can be selected from any $5$ digits.

$=1\times 4\times 5\times 4=80$

$\therefore$ Total$=480+80=560.$